Random Variables and Common Distributions

Manual: General · Subject: Statistics

Model quantitative outcomes with discrete and continuous probability distributions.

Probability Distributions

Random variables

A random variable assigns a numerical value to each outcome of a random process. Its distribution describes the probabilities of possible values, or the density of values in the continuous case.

Common distributions

Distribution	When used	Key parameters
Bernoulli	Single success/failure trial	$p$
Binomial	Count of successes in $n$ independent trials	$n, p$
Poisson	Counts of events over time or space	$\lambda$
Normal	Many natural measurements and sampling averages	$\mu, \sigma$

Random variable properties

Expected value

Long-run average outcome, denoted

E(X)

Variance

Measure of spread around the mean,

Var(X)

Standard deviation

Square root of variance, in original units.

ℹ️

Why the normal distribution appears often

Many sums and averages of small random influences are approximately normal because of the central limit effect.

Which distribution is most appropriate for the number of emails received in an hour if events occur independently at an average rate?

Normal Poisson Uniform Geometric

What do the parameters $n$ and $p$ represent in a binomial distribution?

Matching a distribution to data

1

Step 1: Decide whether the variable is discrete or continuous.
2

Step 2: Identify the process generating the data.
3

Step 3: Check whether trials are repeated, counts are rare, or values are bell-shaped.
4

Step 4: Select the distribution whose assumptions best fit the context.

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Random Variables and Common Distributions

Probability Distributions

Random variables

Matching a distribution to data