Continuous Probability Distributions Reference Cheat Sheet

Key Facts

• For a continuous random variable $X$, the probability at one exact value is $P(X = a) = 0$.
• A probability density function $f(x)$ must satisfy $f(x) \ge 0$ and $\int_{-\infty}^{\infty} f(x)\,dx = 1$.
• The probability that $X$ lies between $a$ and $b$ is $P(a \le X \le b) = \int_a^b f(x)\,dx$.
• The cumulative distribution function is $F(x) = P(X \le x) = \int_{-\infty}^{x} f(t)\,dt$.
• For a continuous random variable, expected value is $E(X) = \int_{-\infty}^{\infty} x f(x)\,dx$.
• Variance is $\operatorname{Var}(X) = E\left(X^2\right) - \left(E(X)\right)^2$, and standard deviation is $\sigma = \sqrt{\operatorname{Var}(X)}$.
• For a uniform distribution on $[a,b]$, $f(x) = \frac{1}{b-a}$, $\mu = \frac{a+b}{2}$, and $\sigma^2 = \frac{(b-a)^2}{12}$.
• For a normal distribution, standardization uses $z = \frac{x - \mu}{\sigma}$ so probabilities can be found from the standard normal model $N(0,1)$.

Vocabulary

Continuous random variable

A variable that can take any value in an interval, such as all real numbers between $2$ and $3$.

Probability density function

A function $f(x)$ whose area over an interval gives the probability that a continuous random variable falls in that interval.

Cumulative distribution function

A function $F(x)$ that gives the probability $P(X \le x)$ for a random variable $X$.

Expected value

The long-run average value of a random variable, calculated for continuous variables by $E(X) = \int_{-\infty}^{\infty} x f(x)\,dx$.

Normal distribution

A bell-shaped continuous distribution described by its mean $\mu$ and standard deviation $\sigma$.

Standard normal distribution

The normal distribution with mean $0$ and standard deviation $1$, written as $N(0,1)$.