Probability Distributions Cheat Sheet

A printable reference covering discrete and continuous distributions, expected value, variance, binomial, normal, and z-scores for grades 10-12.

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Probability distributions describe how likely different outcomes are for a random variable. This cheat sheet helps students compare discrete and continuous models, choose the right formula, and interpret probabilities from tables or graphs. It is useful for homework, tests, and data investigations where probabilities must be calculated and explained. The core ideas are that probabilities must total $1$ for discrete distributions and total area must equal $1$ for continuous distributions. Expected value $\mu = E(X)$ gives the long-run average, while variance $\sigma^2$ and standard deviation $\sigma$ measure spread. Common models such as the binomial and normal distributions use parameters like $n$ , $p$ , $\mu$ , and $\sigma$ to describe shape, center, and variability.

Key Facts

For any probability distribution, $0 \le P(A) \le 1$ and the probabilities of all mutually exclusive outcomes in the sample space add to $1$ .
For a discrete random variable, the mean or expected value is $\mu = E(X) = \sum xP(X=x)$ .
For a discrete random variable, the variance is $\sigma^2 = \sum (x-\mu)^2P(X=x)$ and the standard deviation is $\sigma = \sqrt{\sigma^2}$ .
For a binomial random variable with $n$ trials and success probability $p$ , $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$ for $k=0,1,\dots,n$ .
A binomial distribution has mean $\mu = np$ and standard deviation $\sigma = \sqrt{np(1-p)}$ .
For a continuous random variable, $P(a \le X \le b)$ equals the area under the density curve from $a$ to $b$ , and $P(X=a)=0$ .
A normal random variable is standardized by $z = \frac{x-\mu}{\sigma}$ , which converts $x$ into the number of standard deviations from the mean.
If $X \sim N(\mu,\sigma)$ , then about $68\%$ , $95\%$ , and $99.7\%$ of values lie within $1$ , $2$ , and $3$ standard deviations of $\mu$ .

Vocabulary

Probability distribution: A probability distribution is a model that assigns probabilities to the possible values of a random variable so the total probability is $1$ .
Random variable: A random variable is a variable, usually written as $X$ , whose value depends on the outcome of a chance process.
Probability mass function: A probability mass function gives $P(X=x)$ for each possible value of a discrete random variable.
Probability density function: A probability density function is a curve for a continuous random variable where probabilities are found as areas under the curve.
Expected value: Expected value is the long-run average outcome of a random variable, calculated for discrete data by $E(X)=\sum xP(X=x)$ .
Standard deviation: Standard deviation is a measure of typical distance from the mean, calculated as $\sigma = \sqrt{\sigma^2}$ .

Common Mistakes to Avoid

Adding probabilities to more than $1$ is wrong because a valid probability distribution must have total probability equal to $1$ .
Using the binomial formula when trials are not independent is wrong because $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$ assumes the same $p$ on every trial.
Treating $P(X=a)$ as an area for a continuous distribution is wrong because a single point has no width, so $P(X=a)=0$ .
Confusing variance and standard deviation is wrong because variance is $\sigma^2$ while standard deviation is $\sigma = \sqrt{\sigma^2}$ .
Using $x$ like a z-score is wrong because standard normal probabilities require $z = \frac{x-\mu}{\sigma}$ before using a normal table or calculator.

Practice Questions

1 A discrete random variable has values $0$ , $1$ , and $2$ with probabilities $0.20$ , $0.50$ , and $0.30$ . Find $E(X)$ .
2 Let $X \sim \text{Binomial}(10,0.30)$ . Find $P(X=4)$ using $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$ .
3 A test score is normally distributed with $\mu = 70$ and $\sigma = 8$ . Find the z-score for $x = 86$ and interpret its meaning.
4 Explain whether the number of students absent from a class on a school day is better modeled as a discrete or continuous random variable, and justify your choice.