Continuous Probability Distributions Reference Cheat Sheet

Continuous probability distributions describe random variables that can take any value in an interval, such as time, height, distance, or measurement error. This cheat sheet helps students connect graphs, formulas, and probability statements in a clear way. It is especially useful for distinguishing probability density from actual probability.

Students in grades 11-12 need these tools for statistics, precalculus, calculus, and science applications.

The most important ideas are the probability density function, the cumulative distribution function, expected value, and variance. For a continuous random variable, probabilities come from area under a curve, not from the height of the curve at one point. Common models include the uniform distribution, normal distribution, and exponential distribution.

Standardizing with $z = \frac{x - \mu}{\sigma}$ lets students compare values using the standard normal distribution.

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 $N(0,1)$ .

Common Mistakes to Avoid

Treating $f(a)$ as $P(X = a)$ is wrong because a density height is not a probability. For continuous variables, $P(X = a) = 0$ and probability comes from area.
Forgetting that total area must equal $1$ is wrong because a valid density function must satisfy $\int_{-\infty}^{\infty} f(x)\,dx = 1$ . Always check the full area under the curve.
Using $P(a \le X \le b) = f(b) - f(a)$ is wrong because probability uses the area under $f(x)$ , not the change in height. The correct form is $P(a \le X \le b) = \int_a^b f(x)\,dx$ .
Standardizing with $z = \frac{x + \mu}{\sigma}$ is wrong because the mean must be subtracted. The correct formula is $z = \frac{x - \mu}{\sigma}$ .
Confusing variance and standard deviation is wrong because variance is measured in squared units. Standard deviation is $\sigma = \sqrt{\operatorname{Var}(X)}$ and matches the original units.

Practice Questions

1 A continuous random variable has density $f(x) = \frac{1}{5}$ on $0 \le x \le 5$ . Find $P(1 \le X \le 4)$ .
2 A normal random variable has mean $\mu = 80$ and standard deviation $\sigma = 10$ . Find the $z$ -score for $x = 95$ .
3 For an exponential distribution with rate $\lambda = 0.25$ , use $P(X \le x) = 1 - e^{-\lambda x}$ to find $P(X \le 4)$ .
4 Explain why the probability that a continuous random variable is exactly equal to one specific number is $0$ , even if the density curve is high at that number.

Understanding Continuous Probability Distributions Reference

A density curve is not a picture of individual outcomes. It is a model for how tightly outcomes are packed along a number line. A tall section means values near that location occur more often within small intervals.

Its height can be greater than one when the interval is narrow enough. This is not a problem because probability depends on area. Density has units that depend on the measurement.

If waiting time is measured in minutes, density is measured per minute. Changing from minutes to seconds changes the numerical height of the curve, while the probability of a real time interval stays the same.

The cumulative distribution function is useful because it turns areas into a running total. It starts near zero at very small values and approaches one as values become very large. It never goes down.

A steep part of this graph shows that probability is accumulating quickly there. A flat part shows little or no density. To find the chance of falling between two values, subtract the cumulative total at the lower value from the cumulative total at the upper value.

In calculus, the density is the rate of change of the cumulative function. This connection explains why integration and differentiation appear in probability work.

Expected value is a weighted long run average, not a promise about one trial. A delivery time with an expected value of thirty minutes can still be twenty minutes on one day or fifty on another. Variance describes the spread around the expected value.

Squaring the distances prevents values below the mean from cancelling values above it. Standard deviation returns the spread to the original units, so it is usually easier to interpret.

When comparing two data sets, check both the center and the spread. Two processes can have the same average yet very different reliability.

Each named distribution has assumptions that matter. A uniform model treats every equal length interval in a fixed range as equally likely. It can fit a random position on a fair spinner only when the spinner is designed evenly.

A normal model is symmetric and bell shaped. It often appears when many small independent effects combine, such as measurement errors or some biological traits. Real data can be skewed, limited by zero, or affected by unusual values, so a normal model is not automatic.

An exponential model describes waiting times when events occur at a steady average rate. Its memoryless behavior means that waiting longer does not change the model's remaining waiting time. When solving problems, identify the variable, its units, the possible range, and whether the model assumptions match the situation before using a formula or calculator.

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Continuous Probability Distributions Reference Cheat Sheet

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