What type of events is the Poisson Distribution usually associated with?
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Rare events.
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What type of events is the Poisson Distribution usually associated with?
Rare events.
What are the key characteristics of a Poisson distribution?
It describes events that occur independently and at a constant average rate.
What is the formula for the standard deviation of a Binomial Distribution?
σ = √(np(1 - p)).
What is the probability distribution for an area of 100 cm²?
X ~ Poisson(μ = 2) and Y ~ Poisson(μ = 4).
What is the initial number of tagged fishes captured by the scientist?
250 fishes.
What is a key property of the CDF?
F(x) is an increasing function from 0 to 1.
How do the draws differ when sampling with replacement versus without replacement?
With replacement, the draws are independent; without replacement, the draws are not independent.
What does the variable 'x' represent in the context of the problem?
The number of tagged fishes found in the recapture (16).
What does the notation [n choose x] represent in the Binomial Distribution?
[n choose x] represents the number of ways to choose x successes in n trials, calculated as n! / [x!(n-x)!].
What distribution does the number of heads in 5 tosses of a fair coin follow?
X ~ Binomial(5, 0.5).
What is the relationship between the random variables Y and X when Y = aX + c?
Y is also a discrete random variable.
What is the binomial distribution used for in the given example?
To find the probability of obtaining 0, 1, and 2 heads when tossing a fair coin twice.
What is the sample variance represented by?
s².
If X ~ B(i, p), what does X approximate when the conditions are met?
X can be well-approximated by Y ~ P(μ), where μ = np.
What is the probability Pr(Y ≤ 3) based on the Poisson approximation?
0.8571.
What is the notation used for the number of chocolate drops in each cookie?
X ~ Poisson(μ).
What is the approximate value of e in the limit expression lim n→∞ (1 + 1/n)^n?
The approximate value is 2.718.
How many items are in a lot?
25 items.
What is one of the most widely used discrete distributions?
Poisson Distribution.
When can the Poisson approximation be applied to the Binomial distribution?
When the number of trials is large and the probability of success is small.
What conclusion can the buyer draw from the probability calculation?
It can give him confidence in accepting the lot.
What is an example of a discrete probability distribution involving coin flips?
Flipping a coin 50 times and observing the number of heads.
What is the alternative formula for calculating variance used in the example?
σ² = Σ [ (x - μ)² * Pr(X=x) ]
What is a key property of the CDF?
It is an increasing function from 0 to 1.
How is the maximum probability value for Pr(X = k) determined?
By comparing Pr(X = w + 1) and Pr(X = w).
What happens to the proportion of neutrophils in patients with bacterial or acute viral infections?
It is much higher than 70%.
What does the CDF look like for a continuous random variable?
It is a smooth curve.
What is the mean of the Negative Binomial Distribution?
E(X) = V * p.
What is the goal regarding the probability of chocolate drops in cookies?
To ensure that the probability of each cookie containing at least 1 chocolate drop is greater than 99%.
What does the expression k! in the limit formula signify?
k! is the factorial of k, representing the number of ways to arrange k successes.
If X is the number of colonies in 100 cm², how is X distributed?
X ~ Poisson(μ = 2).
What is the probability Pr(Y ≤ 2) calculated in the example?
0.6767.
What does the random variable X represent in the context of weight?
X = weight of an individual.
What is the probability of obtaining 1 head when tossing a coin twice?
Pr(X = 1) = 1/2.
What kind of events does the Poisson Distribution describe?
Independent events occurring over a period of time or space.
What is the scenario described in the example?
A buyer checks a sample of machine parts from a lot to determine if he should accept it.
What does the 47 Rule of Thumb state about the approximation of random variables?
Pr(X = x) ≈ Pr(Y = x) for all possible values of x.
What distribution does the number of defectives found in the sample follow?
Hypergeometric distribution.
How does the shape of the Poisson distribution change with small μ?
It is heavily skewed to the right.
How is the expected number of successes in n trials calculated?
It is calculated by multiplying the probability of success in one trial (p) by the number of trials (n).
What does the Negative Binomial Distribution model?
The number of times a fixed number of successes occurs in a series of experiments.
How many students are in the STAT1012 midterm exam?
100 students.
What is the formula for the variance of a binomial distribution?
σ² = np(1 - p)
What happens to the CDF as the number of possible values increases?
The CDF will get smoother and smoother.
How do you calculate the probability of no calls in the next minute?
Pr(X = 0).
What is the expected number of bacterial colonies per cm²?
0.02.
What is the mean (μ) for the Poisson approximation in this example?
μ = ip = 2.
What is an example of a random variable when tossing two dice?
X = Sum of the numbers on the two dice.
What is the average number of calls in 6 minutes?
Y ~ Poisson(5 * 2).
What is the formula for E[X²] in a binomial distribution?
E[X²] = Σ (x² * P(X=x)) for x = 0 to n.
What is the probability of no calls in the next 6 minutes?
Pr(Y = 0).
What is the shape of the binomial distribution when p = 0.5?
The distribution is symmetric.
Give an example of a discrete random variable.
The number of heads in a series of coin flips.
How can Poisson probability be calculated rapidly?
By using the recursion relation: Pr(X = x) = μ^x * Pr(X = x - 1), for x = 1, 2, ...
What is a random variable (r.v.)?
A numeric quantity that takes different values with specified probabilities.
What does the variable 'i' represent in the context of the problem?
The number of fishes recaptured (150).
What distribution does the number of recaptured tagged fishes follow?
Hypergeometric distribution.
What is the CDF value for 0 ≤ x < 1?
0.25.
What is the goal for the number of spare calculators to bring for the exam?
To ensure all students have calculators with at least a 90% chance.
In the paper production example, what area is being considered?
An area of 1000m².
What are the parameters of a Binomial distribution?
The number of trials (n) and the probability of success (p).
What is the significance of the empty circle in the CDF?
It indicates that F(1) is excluded from the value 0.129.
How is a random variable X defined in the context of Binomial distribution?
X is the total number of successes in n independent experiments.
What does Pr(X > k) represent in the context of the Geometric Distribution?
It represents the probability of needing more than k trials to achieve the first success.
When does the Negative Binomial Distribution reduce to the Geometric Distribution?
When V = 1.
What condition must be satisfied for Pr(X ≥ 1)?
Pr(X ≥ 1) ≥ 0.99.
What is the mean of a Binomial Distribution?
μ = E[X] = np, where p is the probability of success.
Can discrete random variables take on any value within a range?
No, they can only take specific, distinct values.
What is Euler's constant (w) approximately equal to?
Approximately 2.718.
What are the measures of location for a sample?
Sample Mean, Median, Mode.
What is the variance (σ²) of the random variable X in the example?
0.5.
What is the distribution of the number of white balls drawn without replacement from the same box?
X follows a hypergeometric distribution with N = 30, N1 = 10, N2 = 20, i = 5.
What formula is used to estimate the total number of fishes (N)?
N = (150 * 250) / 16.
What is the formula for the expected value (population mean)?
E(X) = μ.
In the coin flip example, what are we calculating?
The probabilities of obtaining 0, 1, 2, ..., 50 heads.
What is a Binomial Distribution?
A probability distribution that summarizes the likelihood of a value taking on two independent values under a given set of parameters.
What does the Memorylessness property imply about trials in an experiment?
The conditional probability distribution of additional trials does not depend on how many failures have been observed.
What does the Binomial distribution represent?
The probability distribution on the number of successes in independent experiments.
What type of distribution is assumed for the number of chocolate drops in cookies?
Poisson distribution.
What is the probability Pr(Y ≤ 2) based on the Poisson approximation?
0.6767.
What is the distribution of Y, the number of pairs of twins born?
Y follows a Binomial distribution with parameters n = 120 and p = 0.016.
How is the variance of a Binomial Distribution calculated?
σ² = np(1 - p), where n is the number of trials and p is the probability of success.
What is the standard deviation of the Poisson distribution?
σ = μ.
What is the initial number of fishes captured and tagged by the scientist?
250 fishes.
How many of the recaptured fishes were tagged?
16 tagged fishes.
What is a random variable (r.v.)?
A numeric quantity that takes different values with specified probabilities.
What is the condition for p in a Poisson distribution?
p should be small (< 0.11).
What is the conditional probability of getting 6 heads given that there are at least 4 heads when tossing a fair coin 5 times?
Pr(X = 5 | X ≥ 4) = Pr(X = 5) / (Pr(X = 4) + Pr(X = 5)) = 1/6.
What is the probability that a particular student forgot to bring a calculator in the STAT1012 midterm exam?
2%.
What does P(X = x) represent?
The probability that the random variable X takes the value x.
What indicates that the probability is monotonically increasing?
If Pr(X = t + 1) / Pr(X = t) ≥ 1.
How is the expected value of Y calculated?
E(Y) = aE(X) + c.
What is a key characteristic of the Binomial Distribution?
It has a finite number of trials.
What is the formula for the probability of getting exactly k successes in n trials in a Binomial distribution?
The formula is P(X = k) = (n choose k) * p^k * (1 - p)^(n - k).
What is a key characteristic of the Poisson Distribution?
The number of trials can be infinite.
What is the relationship between the random variables Y and X when Y = aX + c?
Y is also a discrete random variable.
What is the probability of obtaining 0 heads when tossing a coin twice?
Pr(X = 0) = 1/4.
What is the probability Pr(Y ≤ 3) calculated in the example?
0.8571.
What does the term 'n choose x' refer to in the pmf of the Binomial distribution?
It refers to the number of ways to choose x successes from n trials.
What are the possible values of x in a Binomial distribution?
x can take values from 0 to n.
What is the mean of the Poisson distribution?
E(X) = μ.
What is the variance of the Poisson distribution?
σ² = μ.
In what scenarios is the Poisson distribution typically applied?
In scenarios like the number of phone calls received at a call center in an hour.
How many fishes did the scientist recapture?
150 fishes.
What is the mean of a Binomial Distribution?
The mean (μ) is calculated as E(X) = np, where n is the number of trials and p is the probability of success.
How many of the recaptured fishes were found to be tagged?
16 tagged fishes.
What is the CDF value for x < 0?
What defines a continuous random variable?
A random variable that can take a continuous range of values over an interval.
What distribution does the number of calls in each minute follow?
X ~ Poisson(5/3).
What is the primary difference between Binomial and Poisson distributions?
Binomial distribution is used for a fixed number of trials with two outcomes, while Poisson distribution is used for counting the number of events in a fixed interval of time or space.
What is the probability of rolling a 1 exactly twice when rolling an unbiased die 7 times?
Pr(X = 2) where X ~ Binomial(7, 1/6).
In a Binomial distribution, what does 'p' represent?
'p' represents the probability of success in a single trial.
What is the conditional probability of the first 2 rolls being a 1 given that a 1 appears exactly twice in 7 rolls?
Pr(Y = 0 | X = 2) where Y ~ Binomial(5, 1/6).
What is the probability mass function (pmf) for the number of neutrophils in a sample of 10?
Pr(X = x) = (10! / (x! (10 - x)!)) * (0.7^x) * (0.3^(10-x)), where x = 0, 1, ..., 10.
What does the variable X represent in the context of tossing a fair coin?
The number of heads among 5 tosses.
What is the probability of observing 10 neutrophils in the sample?
Pr(X = 10) = 0.7^10 = 0.0282.
How is the CDF graphically illustrated?
It looks like a series of steps, called the step function.
How does a probability mass function relate to a frequency distribution?
PMF specifies the population, while frequency distribution summarizes the sample.
What is the shape of the binomial distribution when p > 0.5?
The distribution is left-skewed.
How is the probability mass function (PMF) of a Poisson distribution defined?
P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events.
What is the Poisson approximation used for?
To approximate the Binomial distribution under certain conditions.
What happens to the probability of finding 0 defectives if the number of defectives in the lot increases?
The probability will be even lower.
What is the estimated total number of fishes in the pond?
2343 fishes.
What is the probability of drawing a white ball from the box?
1/3 (since there are 10 white balls out of 30 total).
What is the probability that a particular student forgot to bring a calculator in the STAT1012 midterm exam?
2%.
What is the sample mean represented by?
x̄.
What is the formula to calculate the probability of recapturing 16 tagged fishes?
Pr(X = 16) = (N1 choose 16) * ((N - N1) choose (n - 16)) / (N choose n).
What are the three conditions for approximating a binomial distribution with a Poisson distribution?
i) n ≥ 2000, ii) p < 0.11, iii) np < 5.
What is the probability formula for k successes in the Negative Binomial Distribution?
Pr(X = k) = (k - 1) choose (V - 1) * (1 - p)^(k - V) * p^V.
What is the probability notation for a range of values for a random variable?
P(a < X < b) represents the probability that X falls between a and b.
How do you calculate the probability of at least 2 calls in the next minute?
Pr(X ≥ 2) = 1 - Pr(X = 0) - Pr(X = 1).
What is the formula for the variance of Y?
V(Y) = a²V(X).
What is the distribution of the total number of heads (W = X + Y)?
W ~ B(5, 0.5)
What is the probability we are calculating for the soccer match example?
The probability that the actual scores are 1:0.
What is the probability that the total number of heads (W) equals 0?
Pr(X + Y = 0) = Pr(X = 0) * Pr(Y = 0) = 0.5^5
What is the mean of a Poisson distribution?
The mean of a Poisson distribution is equal to its parameter λ (lambda).
How do you compute Pr(μ - σ ≤ Y ≤ μ + σ)?
Use the normal approximation to the binomial distribution for large n.
What approximation can be used for a binomial distribution with large n and small p?
It can be approximated by a Poisson distribution.
What is the probability of finding 3 bacterial colonies in 100 cm²?
0.1804.
What is the parameter of the Poisson distribution?
The average number of events (λ) in the given interval.
What is the assumption about the number of defective items in the lot?
There are actually 6 defectives in the lot.
What is the measure of location for a population?
Expected Value.
What notation is commonly used for random variables?
Capital letters such as X, Y, Z.
What do x₁, x₂, ..., xᵢ represent in the variance formula?
They are the possible values of the random variable X.
What notation is commonly used for random variables?
Capital letters such as X, Y, Z.
What does E(X) = np < 5 indicate in the context of Poisson distribution?
'Rare events' means only a few events are expected to occur.
How is the Negative Binomial Distribution denoted?
X ~ N(B(V, p)).
What is the extension of the Binomial Distribution formula?
The extension involves calculating E(X) using the formula E(X) = np, where p is the probability of success.
What is the formula for the cumulative distribution function (CDF) of a Geometric Distribution?
F_X(x) = Pr(X ≤ x) = 1 - (1 - p)^x.
If 10 white blood cells are examined, what is the distribution of the number of neutrophils?
X ~ B(10, 0.7).
What are the parameters of the binomial distribution in this example?
i = 2 (number of trials), p = 0.5 (probability of heads).
What condition must be satisfied for the mode to be at i + 1p?
w + 1 ≤ i + 1p.
What is the value of F(0.99999) and F(1.00000)?
F(0.99999) = 0.129 and F(1.00000) = 0.393.
What is the probability Pr(Y ≤ 4) based on the Poisson approximation?
0.9473.
What is the expected value of a Binomial Distribution?
E(X) = n * p.
How can the Poisson Distribution be conceptualized in relation to the Binomial Distribution?
As a Binomial distribution with an infinite number of experiments.
When is the variance of a Binomial Distribution maximized?
When p = 0.5.
What is the binomial distribution with n = 10?
It refers to a binomial distribution with 10 trials.
How is the expected value (μ) mathematically represented?
E(X) = μ = Σ (x_i * Pr(X = x_i)) for i = 1 to n.
What is the probability of finding 0 defectives in the sample?
2.826%.
What is the alternative formula for population variance?
σ² = E[X²] - (E[X])² = Σ (xᵢ² Pr(X = xᵢ)) - (Σ (xᵢ Pr(X = xᵢ)))²
What are the probabilities associated with the outcomes for X?
Pr(X=0) = 1/4, Pr(X=1) = 1/2, Pr(X=2) = 1/4.
How does the mean of the Poisson distribution relate to the Binomial distribution?
The mean of the Poisson distribution is equal to np, where n is the number of trials and p is the probability of success.
Which discrete distribution has the Memorylessness property?
Geometric Distribution.
What distribution does the number of students without a calculator follow?
Binomial distribution, specifically B(100, 0.02).
How is the expected value (E[X]) of a binomial distribution calculated?
E[X] = np
What is the distribution of the number of heads among 1 HKD coins (X)?
X ~ B(2, 0.5)
What are we calculating in the paper production example?
The probabilities of obtaining 0, 1, 2,... holes.
How is the probability of X being greater than k + j expressed?
Pr(X > k + j) = (1 - p)^(k+j).
What is the variance of the Negative Binomial Distribution?
σ² = V * (1 - p) / p².
What is the formula for calculating the probability of obtaining x heads?
Pr(X = x) = i! / (x! (i - x)!) * p^x * (1 - p)^(i - x).
What notation is used to denote a Binomial distribution?
X ~ B(n, p).
What is the probability of finding 0 bacterial colonies in 100 cm²?
0.1353.
How do you calculate the probability of at least 2 calls in the next 6 minutes?
Pr(Y ≥ 2) = 1 - Pr(Y = 0) - Pr(Y = 1).
In the example of tossing two fair coins, what does the random variable X represent?
The number of heads observed.
What is the probability of finding 5 or more colonies in 200 cm²?
Pr(Y ≥ 5) = 1 - Pr(Y ≤ 4) = 0.3711.
What is the Poisson distribution used for?
To model the number of events occurring in a fixed interval of time or space.
What happens to the variance when p = 0 or p = 1?
Variance σ² = 0 due to no uncertainty on X.
What is the expected number of bacterial colonies per cm²?
0.02 per cm².
What is the expected value (E[X]) of the random variable X in the example?
What is the distribution of the number of white balls drawn with replacement from a box of 10 white and 20 black balls?
X follows a Binomial distribution: X ∼ B(5, 1/3).
What is the intuition behind the expected value of 1 when tossing 2 fair coins?
We expect to obtain 1.0 head.
What is the requirement for the number of experiments to observe 'rare events' in a Poisson distribution?
A large n (≥ 2000) is needed.
What is the formula for the expected value E(X) in a Binomial Distribution?
E(X) = Σ (from x=0 to n) [n choose x] p^x (1-p)^(n-x).
What is the average number of calls a telephone operator handles in 3 minutes?
Five calls.
What is the average number of holes in paper production over 100m²?
An average of 2 holes.
What are the two outcomes in a Binomial Distribution?
Success and failure.
How many students are there in the STAT1012 midterm exam?
100 students.
What parameters define a Binomial Distribution?
The number of trials (n) and the probability of success (p).
What is the significance of the parameter μ in the context of the Poisson distribution?
μ represents the mean number of occurrences in a fixed interval for the Poisson distribution.
What is the relationship between E[X²] and variance in a binomial distribution?
Var(X) = E[X²] - (E[X])².
What is the distribution of the number of neutrophils in a sample of 10 white blood cells?
X ~ Binomial(10, 0.7).
In the Poisson Distribution, what happens to the probability as the number of events increases?
The probability becomes very small when the number of events gets larger.
What is the probability mass function (pmf) of the Binomial distribution?
Pr(X = x) = (n choose x) * p^x * (1 - p)^(n - x), for x = 0, 1, 2, ..., n.
What is the probability of observing 10 neutrophils in the sample?
Pr(X = 10) = 0.0282.
What is the probability formula for a Poisson random variable?
Pr(X = k) = (w^(-μ) * μ^k) / k!, where k = 0, 1, 2, ...
What does the parameter μ represent in the Poisson Distribution?
The expected number of events to occur.
How is the Poisson Distribution denoted?
X ~ Poisson(μ).
What is the condition for X2 ~ Binomial(20, 0.1)?
It has marginal n and p, making it an okay approximation.
What is the condition for X3 ~ Binomial(1000, 0.002)?
n is large and p is small, making it a good approximation.
How many fishes did the scientist recapture?
150 fishes.
What is the parameter of the Poisson distribution?
The average rate (λ) of occurrence.
What defines a discrete random variable?
A random variable for which there exists a discrete set of numeric values.
What happens if ip is an integer in the context of the mode?
Then E[X] = ip is the mode.
What is the measure of spread in a population called?
Population Variance.
What is the CDF value for 1 ≤ x < 2?
0.75.
What is the probability Pr(Y ≤ 1) based on the Poisson approximation?
0.4060.
What is the CDF value for x ≥ 2?
What does a probability mass function (PMF) assign to each possible value of a discrete random variable?
A probability.
What is the expected score between teams A and B in soccer matches?
3:2.
What is the expected number of bacterial colonies per 100 cm²?
4 (calculated as 0.02 * 100).
What type of events does the Poisson distribution model?
It models the number of events occurring in a fixed interval of time or space.
What does the variable Y represent in the context of rolling a die?
The number of times a 1 appears among the last 5 rolls.
How does the Poisson distribution relate to the Binomial distribution when n is large and p is small?
The Poisson distribution serves as an approximation for the Binomial distribution under these conditions.
When is it appropriate to use a Poisson distribution instead of a Binomial distribution?
When the number of trials is large and the probability of success is small.
If a ≤ w1(x) ≤ b for all x, what can be inferred about E(w1(X))?
a ≤ E(w1(X)) ≤ b.
What does Pr(1.4 < X < 3.1) represent?
The probability that the random variable X falls between 1.4 and 3.1.
What is the binomial distribution with n = 4?
It refers to a binomial distribution with 4 trials.
What is the significance of discrete random variables in probability?
They are used to model scenarios where outcomes are distinct and countable.
What shape does the Poisson distribution have?
It is a right-skewed distribution.
What does PMF stand for in probability?
Probability Mass Function.
What happens to the Poisson distribution as μ increases?
It becomes more symmetric, though still slightly right-skewed.
In the example of tossing two fair coins, what does the random variable X represent?
The number of heads observed.
What is the definition of the Cumulative Distribution Function (CDF) for a discrete random variable X at value x?
The probability that X is less than or equal to the value x (Notation: Pr(X ≤ x)).
What are the parameters of the Negative Binomial Distribution?
The number of successes (V) and the probability of success (p).
What is the relationship between Binomial and Poisson distributions as n approaches infinity?
As n approaches infinity, the Binomial distribution can be approximated by the Poisson distribution under certain conditions.
What is the goal in estimating the total number of fishes (N)?
To choose N such that Pr(X = 16) is largest.
What is the rule of thumb for approximating a Binomial distribution with a Poisson distribution?
i ≥ 20, p < 0.1, and ip = 2 < 5.
What does the term 'p' represent in the binomial distribution?
The probability of success in a single trial.
What is the distribution of the number of heads among 5 HKD coins (Y)?
Y ~ B(3, 0.5)
What is the proportion of neutrophils in a healthy person's white blood cell count?
70%.
How is the standard deviation of Y related to X?
σ_Y = aσ_X.
What distribution does the number of colonies in 200 cm² follow?
Y ~ Poisson(μ = 4).
How is the expected value of Y calculated?
E(Y) = aE(X) + c.
If w1(x) ≥ 0 for all x, what can be said about E(w1(X))?
E(w1(X)) ≥ 0.
What is the cumulative probability of observing 4 neutrophils?
Pr(X ≤ 4) = 0.0473490.
What is the probability of finding 4 colonies in 200 cm²?
Pr(Y = 4) = e^(-4) * (4^4 / 4!) = 0.1954.
What is the focus of Chapter 3 in STAT 1012?
Discrete Probability Distributions.
What happens to the binomial distribution when p < 0.5?
The distribution is right-skewed.
What is the expected value (population mean) of a random variable X?
The sum of the product of all possible values with their corresponding probabilities.
What is the condition for X1 ~ Binomial(10, 0.2)?
n is too small and p is too large, making it a bad approximation.
What is the Cumulative Distribution Function (CDF) of a discrete random variable X at value x?
The probability that X is less than or equal to the value x (Notation: Pr(X ≤ x)).
What are the parameters of the Binomial distribution?
Number of trials (n) and probability of success (p).
What is the expected value also known as?
Population Mean.
What is the mode of a Binomial distribution?
The mode (k) is the largest integer less than or equal to i + 1p.
What are the parameters for the hypergeometric distribution in this example?
N = 30 (total), N1 = 10 (white), N2 = 20 (black), i = 5 (draws).
How is a random variable defined mathematically?
As a function from a sample space S into the real numbers.
What is the probability of an expectant mother giving birth to twins?
1.6%.
What is the formula for population variance?
V(X) = σ².
What symbol indicates that the distribution of X can be approximated by the Poisson distribution?
The symbol '≈'.
What indicates that the probability is monotonically decreasing?
If Pr(X = t + 1) / Pr(X = t) ≤ 1.
What is the formula for the probability mass function (PMF) of a Binomial Distribution?
P(X = k) = (n choose k) * p^k * (1-p)^(n-k).
How is the probability Pr(X ≤ 1) calculated using the Poisson approximation?
Pr(Y ≤ 1) = e^(-2) * (2^0 / 0!) + e^(-2) * (2^1 / 1!) = 3e^(-2).
What does 'n choose k' represent in the Binomial Distribution formula?
The number of ways to choose k successes in n trials.
What is the probability mass function (pmf) formula for the number of neutrophils?
Pr(X = x) = (10! / (x! (10 - x)!)) * (0.7^x) * (0.3^(10-x)), where x = 0, 1, ..., 10.
What is the variance of a Binomial Distribution?
Var(X) = n * p * (1 - p).
What does Pr(X = 0) represent in this context?
The probability that there are no chocolate drops in a cookie.
What does it mean when p = 0.5 in a binomial distribution?
The distribution is symmetric.
What is the derived condition for μ from the inequality?
μ ≥ ln(100) ≈ 4.6.
What is a discrete random variable?
A variable that can take on a countable number of distinct values.
How many items can the buyer check from the lot?
10 items.
What is the formula for population variance (σ²)?
σ² = E[X - μ²] = Σ (xᵢ - μ)² Pr(X = xᵢ)
What does the variance measure in a population?
The sum of squares of all possible values of X minus the mean (μ) with their corresponding probabilities.
What is the relationship between a sample space S and a random variable X?
A random variable is a function from a sample space S into the real numbers.
What is the property of Memorylessness in probability?
Pr(X > k + j | X > k) = Pr(X > j), where k and j are non-negative integers.
What is the probability that at most 1 student would not bring a calculator in the STAT1012 midterm exam?
Approximately 0.4060.
What is the typical proportion of neutrophils in a healthy person's white blood cell count?
70%.
What are the two types of quantitative variables?
Discrete and continuous variables.
What does the limit expression lim n→∞ (k/n) represent in the context of the Binomial distribution?
It represents the probability p in the Binomial distribution as n approaches infinity.
In the example, how many expectant mothers are in the delivery room?
120 expectant mothers.
What is the minimum number of spare calculators needed to ensure at least a 90% chance that all students have calculators?
4 calculators.
What is the expected number of bacterial colonies in an area of 100 cm²?
What is the notation used for the probability mass function?
Pr(X = x).
What is an example of a random variable when tossing a coin 25 times?
X = Number of heads in 25 tosses.
What is another name for the probability mass function?
Probability distribution.
What is the expectation of a linear combination of functions of X?
E(a w1(X) + b w2(X) + c) = aE(w1(X)) + bE(w2(X)) + c.
What is the probability of finding 1 bacterial colony in 100 cm²?
0.2707.
What is the probability of finding 2 bacterial colonies in 100 cm²?
0.2707.
What is the PMF of the new random variable W = X²?
It is derived from the probability distribution of X by squaring its values.
What is the probability Pr(Y ≤ 1) calculated in the example?
0.4060.
What is the expected value (μ) of Y?
μ = E(Y) = n * p = 120 * 0.016 = 1.92.
What is the exact calculation for Pr(X ≤ 1) using the Binomial distribution?
Pr(X ≤ 1) = (0.98)^100 + 100(0.98)^99(0.02) = 0.4032.
What does 'n' represent in the context of a binomial distribution?
The number of trials.
What is the equivalent condition for Pr(X < 1)?
Pr(X < 1) ≤ 0.01.
What is the probability of obtaining 2 heads when tossing a coin twice?
Pr(X = 2) = 1/4.
In astronomy, what does the Poisson Distribution model?
The number of photons arriving at a telescope.
What is the probability of finding 4 bacterial colonies in 100 cm²?
0.0902.
What is the variance (σ²) of Y?
σ² = V(Y) = n * p * (1 - p) = 120 * 0.016 * (1 - 0.016) ≈ 1.89.
What is the probability of finding 0 colonies in 200 cm²?
Pr(Y = 0) = e^(-4) = 0.0183.
Give an example of a real-life application of the Poisson Distribution in electrical systems.
The number of telephone calls arriving.
What are the possible outcomes when tossing two fair coins?
{HH, HT, TH, TT}.
What is the probability of finding 3 colonies in 200 cm²?
Pr(Y = 3) = e^(-4) * (4^3 / 3!) = 0.1954.
How do you compute the variance of Y if Y = 1 - 4X?
Use the formula V(Y) = 16V(X).
What could be the implications of observing all neutrophils in the sample?
The person is either healthy but unlucky or has a viral/bacterial infection (p >> 0.7).
What is the significance of the term '1 - p' in the variance formula?
It represents the probability of failure in a single trial.
What could it indicate if a person has a high proportion of neutrophils?
The person may have a viral or bacterial infection.
What inequality must hold for Pr(X = 0)?
Pr(X = 0) = e^(-μ) ≤ 0.01.
What is the probability of getting 0 heads when tossing two fair coins?
Pr(X = 0) = 1/4.
How do you calculate the probability that W equals 1?
Pr(X + Y = 1) = Pr(X = 0, Y = 1) + Pr(X = 1, Y = 0) = 5 * 0.5^5
What is the probability Pr(Y ≤ 4) calculated in the example?
0.9473.
How does the probability change as the number of trials increases?
The probabilities for k successes become more spread out and can be calculated using the binomial formula.
What is the cumulative distribution function (CDF) of X?
It is a function that shows the probability that X will take a value less than or equal to x.
What is the formula for the variance of Y?
V(Y) = a²V(X).
How is the standard deviation of Y related to X?
σ_Y = aσ_X.
What is the first step in computing the expected value E(X)?
Identify the probability distribution of X.
What does a probability of Pr(X = 0) = 0.0000059 suggest?
It suggests that having no neutrophils is extremely unlikely.
What is the property of the sum of two independent binomial random variables?
If X ~ B(n1, p) and Y ~ B(n2, p), then X + Y ~ B(n1 + n2, p).
What is the question regarding the difference of the two random variables (X - Y)?
The distribution of X - Y is not straightforward and requires further analysis.
What is the probability of getting 2 heads when tossing two fair coins?
Pr(X = 2) = 1/4.
How can the Binomial Distribution be applied in a coin flipping scenario?
By flipping a coin n times.
What is the probability of finding 1 colony in 200 cm²?
Pr(Y = 1) = e^(-4) * (4^1 / 1!) = 0.0733.
What is the probability of finding 2 colonies in 200 cm²?
Pr(Y = 2) = e^(-4) * (4^2 / 2!) = 0.1465.
How is the Poisson Distribution used in biology?
To model the number of mutations on DNA per unit time.
In biology, how is the Poisson Distribution relevant to white blood cells?
It models the number of neutrophils out of n white blood cells.
If w1(x) ≥ w2(x) for all x, what can be concluded about their expectations?
E(w1(X)) ≥ E(w2(X)).
What is a financial application of the Poisson Distribution?
The number of losses or claims occurring in a given period of time.
What is the cumulative probability of observing 6 neutrophils?
Pr(X ≤ 6) = 0.3503893.
What is the probability of finding 5 or more bacterial colonies in 100 cm²?
0.0527.
