Continuous Probability Distribution

Describes a continuous random variable.

Probability Density Function

Denoted by $f$ . Does not give the probability at an exact point (which is always 0 for continuous variables). Instead, gives the relative likelihood of the random variable taking on a value in a small interval around a point.

The probability of an event is the integral of the PDF over the region corresponding to the event:

P(a \leq X \leq b) = \int_{a}^{b} f(x) dx

Probability of all the values combined, is 1:

\int_{-\infty}^{\infty} f(x)\; \text{d}x = 1

Mean

The integral is taken over the entire domain of the random variable

E(X) = \mu = \int_{-\infty}^{\infty} x \cdot f(x) \, dx

Here $f(x)$ is the probability density function.

Variance

\text{Var}(X) = \sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2 \cdot f(x) \, dx

An equivalent computational formula is:

\text{Var}(X) = \int_{-\infty}^{\infty} x^2 \cdot f(x) \, dx - \mu^2

Covariance

\text{Cov}(X,Y) = E(XY) - E(X)E(Y)

Cumulative Distribution Function

F(x) = \int_{-\infty}^{x} f(t) dt

Example

Consider a continuous random variable $X$ that follows a uniform distribution over the interval [0, 4], with probability density function:

f(x) = \begin{cases} 0.25 & \text{for } 0 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}

Solution Reveal Hide

The mean would be: $E(X) = \int_{0}^{4} x \cdot \frac{1}{4} \, dx = \frac{1}{4} \cdot \frac{x^2}{2} \bigg|_{0}^{4} = \frac{1}{4} \cdot \frac{16}{2} = 2$ .

The variance would be: $\text{Var}(X) = \int_{0}^{4} (x-2)^2 \cdot \frac{1}{4} \, dx = \frac{1}{4} \cdot \frac{16}{12} = \frac{4}{3} \approx 1.33$ .

Types

Uniform Distribution

A type of continuous probability distribution where all outcomes are equally likely within a specified range. Has the PDF:

f(x) = \begin{cases} \frac{1}{b-a} & \text{for } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}

The mean, variance and CDF are:

E(X) = \mu = \frac{a + b}{2}

\text{Var}(X) = \sigma^2 = \frac{(b-a)^2}{12}

F(x) = \begin{cases} 0 & \text{for } x < a \\ \frac{x-a}{b-a} & \text{for } a \leq x \leq b \\ 1 & \text{for } x > b \end{cases}

Normal Distribution

Describes data that clusters around a mean value, forming a symmetric bell-shaped curve. Denoted by $N(\mu, \sigma^2)$ . Has PDF:

f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp{\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)}

Follows the empirical rule.

It’s often used to model real-world phenomena like heights, test scores, or measurement errors, where most values are near the mean, and fewer occur as you move away from it.

Standard Normal Distribution

A special case of the normal distribution where $\mu=0$ and $\sigma= 1$ . Denoted by $N(0, 1)$ . Has PDF:

f(x) = \frac{1}{\sqrt{2\pi}} \exp{\left(-\frac{x^2}{2}\right)}

Chi-Square Distribution

Distribution of the sum of the squares of $k$ independent standard normal random variables, where $k$ is the degrees of freedom. Has the PDF:

f(x; k) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, \quad x > 0

Here:

$k$: the degrees of freedom
$\Gamma$: the gamma function

Used in tests like the Chi-square goodness-of-fit test and tests for independence in contingency tables.

Student’s t-Distribution

A more conservative form of the standard normal distribution. Has heavier tails than the standard normal distribution. Denoted by $X \sim t(v)$ . Here the parameter $v$ is the degrees of freedom, and is often $v=n-1$ . Gets close to standard normal distribution as $v \to \infty$ . Centered at zero.

For a Student’s t-distribution, with $v \gt 2$ :

\text{Var}(X) = \frac{v}{v-2}

Used when:

Sample size $n < 30$
Population standard deviation is unknown

Noncentral t-Distribution

A generalization of the Student’s t-distribution. Denoted as $X \sim t(v,\delta)$ . Centered at $\delta$ .