What is the variance Var[X] of a discrete uniform distribution with outcomes from a to b?
Var[X] = ((b - a + 1)^2 - 1) / 12.
What is the formula for the probability density function (PDF) of a continuous uniform distribution?
f(x) = 1 / (b - a) for a ≤ x ≤ b.
What is the expected value E[X] of a continuous uniform distribution with range from a to b?
E[X] = (a + b) / 2.
What is the variance Var[X] of a continuous uniform distribution with range from a to b?
Var[X] = (b - a)^2 / 12.
What is the formula for the probability of getting exactly k successes in n trials in a binomial distribution?
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Fill in the blank: The expected value E[X] for a binomial distribution is _____ .
n * p
What is the variance Var[X] for a binomial distribution?
Var[X] = n * p * (1 - p)
What happens to the binomial distribution as n increases and p remains constant?
It approaches a normal distribution.
What is the expected value E[X] of a Poisson distribution?
E[X] = λ
What is the variance var[X] of a Poisson distribution?
var[X] = λ
Which of the following is a property of the Poisson distribution?
The sum of independent Poisson random variables is also Poisson distributed.
What is the relationship between the Poisson distribution and the exponential distribution?
The time between events in a Poisson process follows an exponential distribution.
What happens to the Poisson distribution as λ approaches infinity?
It approaches a normal distribution.
What is a key assumption of the Poisson distribution?
Events occur independently of each other.
Fill in the blank: For small values of λ, the Poisson distribution approximates the _____ distribution.
Binomial
What is the probability mass function (PMF) of a geometric distribution?
The PMF of a geometric distribution is given by P(X = k) = (1 - p)^(k - 1) * p, where p is the probability of success.
What is E[X] for a geometric distribution?
E[X] = 1/p, where p is the probability of success.
What is Var[X] for a geometric distribution?
Var[X] = (1 - p) / p^2.
What is the PMF of a negative binomial distribution?
The PMF is given by P(X = k) = (k - 1) choose (r - 1) * p^r * (1 - p)^(k - r), where r is the number of successes and p is the probability of success.
What is E[X] for a negative binomial distribution?
E[X] = r / p, where r is the number of successes and p is the probability of success.
What is Var[X] for a negative binomial distribution?
Var[X] = r(1 - p) / p^2.
Fill in the blank: The geometric distribution is a special case of the __________ distribution.
negative binomial distribution.
Why is Normal distribution popular?
Because of Central Limit Theorem
Properties of Normal Distribution
• Symmetric around mean μ.
• Skewness = 0, Kurtosis = 3.
• Linear combination of independent normal random variables is also normal.
• 68% of values within 1σ, 95% within 2σ, 99.7% within 3σ.
