Continuous Probability Distributions: Continuous Random Variables and Their Properties

Definition

A continuous random variable is a random variable that can take any real value within a given interval or collection of intervals. Its probabilities are described by a probability density function (pdf), denoted by $f(x)$, such that:

• $f(x) \ge 0$ for all $x$
• The total area under the curve of $f(x)$ over its entire range is 1
• The probability that $X$ lies between two values $a$ and $b$ is given by the area under the curve from $a$ to $b$:

$$P(a \le X \le b) = \int_a^b f(x)\,dx$$

For a continuous random variable:

$$P(X = c) = 0$$

for any exact value $c$.

A continuous probability distribution is the probability model defined by such a random variable and its density function.

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Main Content

1. Continuous Random Variables and Probability Density Function

• A continuous random variable can assume infinitely many values in an interval, such as all real numbers between 0 and 10, or all positive values.
• Its behavior is described by a probability density function, which is not a probability itself but a function whose area over an interval gives the probability.

Understanding the probability density function

The pdf $f(x)$ must satisfy:

1. Non-negativity

• : $$f(x) \ge 0$$ for every $x$.

2. Total area equals 1

• : $$\int_{-\infty}^{\infty} f(x)\,dx = 1$$

3. Interval probability

• : $$P(a \le X \le b) = \int_a^b f(x)\,dx$$

The pdf tells us where values are more likely to occur, but not the probability of a specific point. The height of the graph of $f(x)$ does not directly represent probability; the area under the curve does.

Example

Suppose $X$ has pdf:

$$f(x) = \begin{cases} 2x, & 0 \le x \le 1 \\ 0, & \text{otherwise} \end{cases}$$

Then:

$$\int_0^1 2x\,dx = \left[x^2\right]_0^1 = 1$$

so it is a valid pdf.

The probability that $X$ lies between 0.2 and 0.8 is:

$$P(0.2 \le X \le 0.8) = \int_{0.2}^{0.8} 2x\,dx = \left[x^2\right]_{0.2}^{0.8} = 0.64 - 0.04 = 0.60$$

Key idea

For continuous variables, probabilities are obtained from areas, not from direct values at points.

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2. Cumulative Distribution Function and Core Properties

• The cumulative distribution function (cdf) $F(x)$ gives the probability that the random variable is less than or equal to $x$: $$F(x) = P(X \le x)$$

• The cdf is related to the pdf by: $$F(x) = \int_{-\infty}^{x} f(t)\,dt$$ and, when differentiable, $$f(x) = \frac{d}{dx}F(x)$$

Properties of the cdf

The cumulative distribution function has important properties:

1. Monotonic non-decreasing

As $x$ increases, $F(x)$ never decreases.

2. Limits at infinity

$$\lim_{x\to -\infty}F(x)=0,\qquad \lim_{x\to\infty}F(x)=1$$

3. Probability on an interval

$$P(a < X \le b)=F(b)-F(a)$$

4. No jump at exact points for continuous variables

Since exact probabilities are zero, $$P(X=c)=0$$

Example

If $F(x)$ is given by:

$$F(x)= \begin{cases} 0, & x<0 \\ x^2, & 0\le x\le 1 \\ 1, & x>1 \end{cases}$$

then:

• $P(X\le 0.5)=F(0.5)=0.25$
• $P(0.2<X\le0.8)=F(0.8)-F(0.2)=0.64-0.04=0.60$

Visual intuition

Probability is the accumulated area under the density curve:

f(x)
 ^
 |                 .
 |               .   .
 |             .       .
 |           .           .
 |_________._______________.________> x
          a               b

Area between a and b = P(a <= X <= b)

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3. Expected Value, Variance, and Important Distributions

• The expected value or mean of a continuous random variable describes its long-run average or center: $$E(X)=\mu=\int_{-\infty}^{\infty} x f(x)\,dx$$

• The variance measures spread around the mean: $$\operatorname{Var}(X)=E[(X-\mu)^2]=\int_{-\infty}^{\infty}(x-\mu)^2 f(x)\,dx$$ and can also be written as: $$\operatorname{Var}(X)=E(X^2)-[E(X)]^2$$

Properties of continuous random variables

Some fundamental properties include:

1. Mean as balance point

The mean indicates the central tendency of the distribution.

2. Variance and standard deviation

Standard deviation is the square root of variance: $$\sigma=\sqrt{\operatorname{Var}(X)}$$ It measures dispersion in the same units as the variable.

3. Linearity of expectation

For constants $a,b$: $$E(aX+b)=aE(X)+b$$

4. Transformation of variance

$$\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$$

Example

For the pdf: $$f(x)=2x,\quad 0\le x\le 1$$

Mean: $$E(X)=\int_0^1 x(2x)\,dx=2\int_0^1 x^2\,dx=2\cdot\frac{1}{3}=\frac{2}{3}$$

Second moment: $$E(X^2)=\int_0^1 x^2(2x)\,dx=2\int_0^1 x^3\,dx=2\cdot\frac{1}{4}=\frac{1}{2}$$

Variance: $$\operatorname{Var}(X)=\frac{1}{2}-\left(\frac{2}{3}\right)^2 =\frac{1}{2}-\frac{4}{9} =\frac{1}{18}$$

Standard deviation: $$\sigma=\sqrt{\frac{1}{18}}$$

Common continuous distributions

Continuous distributions are widely used in statistics and probability. Some standard ones are:

Uniform distribution

• : all values in an interval are equally likely

Normal distribution

• : bell-shaped, used in natural and social sciences

Exponential distribution

• : models waiting times

Gamma, Beta, and Chi-square distributions

• : used in advanced statistical modeling

For example, the normal distribution is important because many real-world variables approximately follow it, and because of the Central Limit Theorem.

Normal distribution shape

density
  ^
  |                   *
  |                *     *
  |              *         *
  |            *             *
  |          *                 *
  |_________*_________*_________*______> x
            μ-σ       μ       μ+σ

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Working / Process

1. Identify the random variable

• Determine whether the variable is continuous.
• Check whether the outcomes lie on a scale or interval, such as time, weight, length, or temperature.

2. Write the probability model

• Find or use the probability density function $f(x)$.
• Verify that $f(x) \ge 0$ and that the total area under the curve equals 1.

3. Compute probabilities and measures

• Use integration to find interval probabilities: $$P(a \le X \le b)=\int_a^b f(x)\,dx$$

• Use the pdf to compute the mean, variance, and other measures: $$E(X)=\int_{-\infty}^{\infty}x f(x)\,dx$$ $$\operatorname{Var}(X)=\int_{-\infty}^{\infty}(x-\mu)^2 f(x)\,dx$$

Example process

Suppose a variable has pdf: $$f(x)= \begin{cases} 1, & 0\le x\le 1 \\ 0, & \text{otherwise} \end{cases}$$

This is a uniform distribution on $[0,1]$.

• Step 1: Verify the pdf: $$\int_0^1 1\,dx=1$$

• Step 2: Find probability: $$P(0.3\le X\le 0.7)=\int_{0.3}^{0.7}1\,dx=0.4$$

• Step 3: Find mean: $$E(X)=\int_0^1 x\,dx=\frac{1}{2}$$

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Advantages / Applications

• Continuous probability distributions are essential for modeling real-world measurements that vary smoothly, such as height, rainfall, speed, and response time.
• They provide a mathematical foundation for statistical inference, hypothesis testing, estimation, and confidence intervals.
• They are widely used in engineering, physics, economics, biology, medicine, and quality control to model uncertainty and variability.

Additional applications

Reliability engineering

• : modeling time to failure of machines

Queueing theory

• : studying waiting times in service systems

Finance

• : modeling asset returns and risk

Health sciences

• : analyzing blood pressure, drug concentration, and survival times

Measurement systems

• : handling experimental errors and noise

Why they are useful

Continuous distributions allow analysts to:

• summarize random behavior with a compact mathematical model
• compute probabilities over intervals
• predict outcomes and make decisions under uncertainty
• compare theoretical models with observed data

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Summary

• Continuous random variables can take infinitely many values in an interval, and probabilities are assigned through areas under a density curve.
• The pdf and cdf are the two main functions used to describe continuous distributions.

• Mean, variance, and standard deviation are the key measures used to summarize center and spread.

• Important terms to remember: continuous random variable, probability density function, cumulative distribution function, expected value, variance, standard deviation