MAT 521: Probability Distributions and Expectations

What is notation for Bernoulli

X ~ Bern(p), P(X = 1) = p, P(X = 0) = 1 - p

What is the notation for a Binomial distribution?

X ~ Bin(n, p)

What is the probability mass function (PMF) for a Binomial distribution?

P(X = k) = (n choose k) p^k (1 - p)^(n - k)

What are the possible values of k in a Binomial distribution?

k = 0, ..., n

What is the expected value E[X] of a Binomial distribution?

E[X] = np

What is the distribution represented by X ~ HyperGeo(w, b, n)?

Hypergeometric distribution

What is the PMF formula for a Hypergeometric distribution?

P(X = k) = [(w choose k)(b choose n - k)] / (w + b choose n)

What are the possible values of k in a Hypergeometric distribution?

k = 0, ..., n

What is the distribution type for a Discrete Uniform random variable?

X ~ Unif(C)

What is the PMF for a Discrete Uniform random variable?

P(X = t) = 1 / |C| if t ∈ C

How is the PMF calculated for a subset A of C in a Discrete Uniform distribution?

P(X = A) = |A| / |C|

What is the expected value E[X] for a Discrete Uniform random variable?

E[X] = (a + b) / 2

What distribution is represented by X ~ Geom(p)?

Geometric distribution

What does X represent in a geometric distribution?

The number of failures before the 1st success

What is the PMF of a geometric distribution?

P(X = k) = (q)^k * p, where q = 1 - p

What is the expected value E[X] of a geometric distribution?

E[X] = q / p

What is the variance Var[X] of a geometric distribution?

Var[X] = q / p^2

What is the distribution of First Success

X ~ Fs.(p)

What does First Success measure

number of trials until 1st success

What is Expected Value for First Success

E[X] = 1 / p

What is PMF for First Success

PMF: P(X = k) = (1 - p)^(k - 1) * p, k = 1, 2, ...

What is the distribution of a Negative Binomial random variable?

X ~ NegBin(r, p)

What does the Negative Binomial distribution represent?

The number of failures before the rth success.

What is the PMF of a Negative Binomial distribution?

P(X = k) = (n + r - 1 choose r - 1) p^r (1 - p)^k

What is the expected value E[X] of a Negative Binomial distribution?

E[X] = r(1 - p)/p

What is the variance Var[X] of a Negative Binomial distribution?

Var[X] = r(1 - p)/p^2

What is the distribution for the number of failures before the rth success in a negative hypergeometric distribution?

X ~ NHGeo(m, N, r)

In a negative hypergeometric distribution, what does N represent?

Total population size

In a negative hypergeometric distribution, what does K represent?

# successes in the population

In a negative hypergeometric distribution, what does r represent?

# successes waiting for

What is the expected value E[X] in a negative hypergeometric distribution?

E[X] = r(N - K) / (K + 1)

What is the distribution of a random variable X that follows a Poisson distribution?

X ~ Pois(λ)

What is the probability mass function (PMF) of a Poisson random variable?

P(X = k) = e^(-λ) λ^k / k!

What is the expected value E[X] of a Poisson random variable?

E[X] = λ

What is the variance Var[X] of a Poisson random variable?

Var[X] = λ

What does a Poisson random variable X count?

The number of successes in a fixed region

What does λ represent in a Poisson distribution?

The rate of occurrence

What is the distribution of a uniform random variable?

X ~ Unif(a, b)

What is the expected value E[X] of a uniform random variable X ~ Unif(a, b)?

E[X] = (a + b) / 2

What is the variance Var[X] of a uniform random variable X ~ Unif(a, b)?

Var[X] = (b - a)^2 / 12

What is the notation for a normal distribution?

X ~ N(μ, σ²)

What does μ represent in a normal distribution?

Mean

What does σ² represent in a normal distribution?

Variance

What is the probability density function (PDF) of a normal distribution?

f(x) = φ((x - μ) / σ)

What is the cumulative distribution function (CDF) of a normal distribution?

F(x) = ((x - μ) / σ(1/ σ))

What is the distribution of a random variable X that models time between independent events occurring at a constant average rate?

X ~ Expo(λ)

What does λ represent in the exponential distribution?

The rate of events per unit of time

What is the probability density function (PDF) of the exponential distribution?

f(x) = λ e^(-λx)

What is the cumulative distribution function (CDF) of the exponential distribution?

F(x) = 1 - e^(-λx)

What is the expected value E[X] of a random variable X following an exponential distribution?

E[X] = 1 / λ

What is the variance Var[X] of a random variable X following an exponential distribution?

Var[X] = 1 / λ²