Topic 2 of 7

Discrete and continuous random variables

Step 22 · Probability theory topics

Describe random variables by their probability mass or density functions, compute probabilities from them, and use cumulative distribution functions for both kinds.

Probability mass functions

A discrete random variable takes separate values, like counts. Its probability mass function p(x) = P(X = x) gives each value's probability. The probabilities are never negative, and they add to 1.

Distribution	p(k)	Mean
Bernoulli(p)	p for 1, 1 − p for 0	p
Binomial(n, p)	$\binom{n}{k}p^{k}(1 - p)^{n-k}$	np
Geometric(p)	$(1 - p)^{k-1}p$	$\frac{1}{p}$
Poisson(λ)	$\frac{e^{-\lambda}\lambda^{k}}{k!}$	λ

Example: Geometric

A free throw goes in with probability 0.7. What is the probability the first miss comes on the third throw?

Make, make, then miss: 0.7 × 0.7 × 0.3.

Answer0.147

Example: Poisson

A help desk gets 3 calls an hour on average. What is the probability of no calls in an hour?

For a Poisson(3) count, $P(X = 0) = e^{-3}$ .

AnswerAbout 0.050

Probability density functions

A continuous random variable takes values across a whole interval. Its density f(x) gives probabilities as areas: P(a ≤ X ≤ b) is the integral of f from a to b. The total area is 1, and any single value has probability 0.

P(a \le X \le b) = \int_{a}^{b} f(x)\,dx \qquad \int_{-\infty}^{\infty} f(x)\,dx = 1

Example: A uniform variable

X is uniform on [0, 4]. What is P(1 ≤ X ≤ 2.5)?

The density is 1/4 on [0, 4].
The area from 1 to 2.5 is 1.5 × 1/4.

Answer0.375

Example: An exponential variable

A bulb's lifetime in years has density $f(x) = 0.5e^{-0.5x}$ for x ≥ 0. What is P(X > 2)?

$\int_{2}^{\infty} 0.5e^{-0.5x}\,dx = e^{-1}$ .

AnswerAbout 0.368

An exponential density, drawn 10 times taller so its shape shows. The shaded area beyond x = 2 is the chance a bulb lasts more than 2 years, about 0.37.

Cumulative distribution functions

The cumulative distribution function F(x) = P(X ≤ x) works for both kinds. It rises from 0 to 1. For a continuous variable it is the integral of the density, and the density is its derivative.

Example: Uniform

Find F(x) for X uniform on [0, 4].

For 0 ≤ x ≤ 4, F(x) is the area up to x, which is x/4.
F is 0 below 0 and 1 above 4.

AnswerF(x) = x/4 on [0, 4]

Example: From F to probabilities

$F(x) = 1 - e^{-x}$ for x ≥ 0. Find P(1 < X ≤ 2).

P(1 < X ≤ 2) = F(2) − F(1).
That is $(1 - e^{-2}) - (1 - e^{-1}) = e^{-1} - e^{-2}$ .

AnswerAbout 0.233

Try it yourself

Work each one on paper first, then open the answer to check.

X is 1, 2 or 3 with probabilities 0.2, 0.5 and c. What is c?
Show answer
0.3
X is binomial with n = 4 and p = 0.5. What is P(X = 4)?
Show answer
$\frac{1}{16}$
X is uniform on [2, 6]. What is P(X ≤ 3)?
Show answer
0.25
f(x) = 2x on [0, 1]. What is P(X ≤ 0.5)?
Show answer
0.25
X is continuous. What is P(X = 3)?
Show answer
0
F(5) = 0.8 and F(2) = 0.3. What is P(2 < X ≤ 5)?
Show answer
0.5