Math 183 - Part 4 (Probability Distributions)

Bernoulli distribution

A distribution with two possible outcomes (1 and 0), where P(X=1) = p and P(X=0) = 1−p.

Bernoulli expectation

The mean of a Bernoulli(p) variable is E(X) = p.

Bernoulli variance

The variance of a Bernoulli(p) variable is Var(X) = p(1−p).

Binomial distribution

The distribution of the number of “successes” in n independent Bernoulli(p) trials; PMF: P(Y=y) = C(n,y) p^y (1−p)^{n−y}.

Binomial expectation

The mean of a Binomial(n,p) variable is E(Y) = np.

Binomial variance

The variance of a Binomial(n,p) variable is Var(Y) = np(1−p).

Geometric distribution

The distribution of the number of independent Bernoulli(p) trials until the first “success”; PMF: P(Y=y) = (1−p)^{y−1}p for y ≥ 1.

Geometric expectation

The mean of a Geometric(p) variable is E(Y) = 1/p.

Geometric variance

The variance of a Geometric(p) variable is Var(Y) = (1−p)/p^2.

Negative binomial distribution

The distribution of the number of trials until m successes in independent Bernoulli(p) trials; PMF: P(Y=y) = C(y−1, m−1) p^m (1−p)^{y−m} for y ≥ m.

Negative binomial expectation

The mean of a Negative Binomial(m,p) variable is E(Y) = m/p.

Negative binomial variance

The variance of a Negative Binomial(m,p) variable is Var(Y) = m(1−p)/p^2.

Poisson distribution

A distribution modeling the number of rare events in a fixed interval; PMF: P(Y=y) = e^(−λ) λ^y / y! for y ≥ 0.

Poisson expectation

The mean of a Poisson(λ) variable is E(Y) = λ.

Poisson variance

The variance of a Poisson(λ) variable is Var(Y) = λ.

Poisson approximation

The Poisson(λ = np) distribution approximates Binomial(n, p) when n is large and p is small.

Uniform distribution (continuous)

A variable that is equally likely to take any value in the interval [a, b]; PDF: f(x) = 1/(b−a) for x in [a, b].

Uniform expectation

The mean of a Uniform(a, b) variable is E(X) = (a+b)/2.

Uniform variance

The variance of a Uniform(a, b) variable is Var(X) = (b−a)^2/12.

Exponential distribution

Models the waiting time between Poisson events; PDF: f(x) = λ e^(−λx) for x ≥ 0.

Exponential expectation

The mean of an Exponential(λ) variable is E(X) = 1/λ.

Exponential variance

The variance of an Exponential(λ) variable is Var(X) = 1/λ^2.

Normal distribution

A continuous, bell-shaped distribution determined by mean μ and standard deviation σ; PDF: f(y) = exp(−(y−μ)^2/(2σ^2)) / sqrt(2πσ^2).

Standard normal distribution

The normal distribution with μ = 0 and σ = 1.

Central Limit Theorem (CLT)

The sum (or average) of many independent, identically distributed random variables approaches a normal distribution as sample size increases.

Normal approximation to binomial

For large n, Binomial(n, p) ≈ Normal(np, np(1−p)).

Continuity correction

A technique to improve normal approximations for discrete distributions by adding or subtracting 0.5 to the cutoff value.

Chi-square distribution

The sum of squares of m independent standard normal random variables; used in variance estimation and goodness-of-fit tests.

t-distribution

Distribution of the ratio of a standard normal variable and the square root of an independent chi-square variable divided by its degrees of freedom; used for inference with small samples.

F-distribution

The ratio of two independent chi-square variables (each divided by their respective degrees of freedom); used in comparing variances (ANOVA).

Simulation (in probability)

Using pseudo-random number generators to produce realizations from probability distributions for empirical exploration.

Law of large numbers (simulations)

Simulated averages and proportions converge to theoretical values as the number of trials increases.

Bernoulli example

Flipping a coin once; “success” = heads.

Binomial example

Number of heads in 10 coin flips.

Geometric example

Number of flips until the first heads appears.

Negative binomial example

Number of trials needed to get the third heads in repeated coin flips.

Poisson example

Number of rare events, like radioactive decays, in a given time interval.

Uniform example

Randomly selecting a number from a line segment.

Exponential example

Time between bus arrivals if arrivals are random (Poisson process).

Normal example

Heights, IQ scores, and other naturally varying phenomena.

Poisson process

A process where events happen independently at a constant average rate; interarrival times are exponentially distributed.