Probability for MATH 141

Understanding Probability with Integration

You've seen how integrals can compute areas, volumes, and averages. Now we'll use integration for probability — answering questions like "What's the chance that a random value falls between 2 and 5?"

In calculus-based probability, we work with continuous random variables — quantities that can take any value in a range (like heights, waiting times, or temperatures), not just whole numbers.

Probability Density Functions (PDFs)

A probability density function $f(x)$ describes how probability is distributed across values. It must satisfy two conditions:

1. $f(x) \geq 0$ for all $x$ (probabilities can't be negative)
2. $\int_{-\infty}^{\infty} f(x) \, dx = 1$ (total probability is 100%)

Key insight: The PDF itself is NOT a probability. The area under the curve gives probability!

Finding Probability

The probability that a random variable $X$ falls between $a$ and $b$ is:

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

This is just the area under the PDF curve from $a$ to $b$.

Expected Value (Mean)

The expected value (or mean) of a continuous random variable is:

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

This is the "weighted average" of all possible values, weighted by their probabilities. Think of it as the "balance point" of the PDF.

Variance and Standard Deviation

The variance measures how spread out the distribution is:

$$\text{Var}(X) = E(X^2) - [E(X)]^2$$

where $E(X^2) = \int_{-\infty}^{\infty} x^2 \cdot f(x) \, dx$

The standard deviation is $\sigma = \sqrt{\text{Var}(X)}$.

What Does Variance Mean Intuitively?

Low variance = values cluster tightly around the mean.

Example: Heights of adult men in the US have a mean around 5'9" with standard deviation ~3 inches. Most men are between 5'6" and 6'0". The distribution is "tight."

High variance = values are spread out widely.

Example: Annual incomes have a mean around \$60,000, but the standard deviation is huge (~\$50,000+). Some people earn \$20,000, others earn \$500,000. The distribution is "spread out."

The Exponential Distribution

A common PDF is the exponential distribution:

$$f(x) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0$$

This models waiting times (time until the next event).

• Mean: $E(X) = \frac{1}{\lambda}$
• Variance: $\text{Var}(X) = \frac{1}{\lambda^2}$

Common Mistakes and Misunderstandings

❌ Mistake: Thinking $f(x)$ is a probability

Wrong: "The probability at $x = 2$ is $f(2) = 0.3$."

Why it's wrong: For continuous distributions, the probability of any exact value is zero! The PDF gives probability density, not probability.

Correct: $P(a \leq X \leq b) = \int_a^b f(x) \, dx$. Only the area gives probability.

❌ Mistake: Forgetting that total probability must equal 1

Wrong: Using $f(x) = x^2$ on $[0, 2]$ without checking if it integrates to 1.

Why it's wrong: $\int_0^2 x^2 \, dx = \frac{8}{3} \neq 1$, so this isn't a valid PDF!

Correct: Always verify $\int f(x) \, dx = 1$, or find the normalizing constant $c$.

❌ Mistake: Using wrong limits for expected value

Wrong: $E(X) = \int_0^{\infty} x \cdot f(x) \, dx$ when the PDF is only defined on $[0, 1]$.

Why it's wrong: You must integrate over the support of the PDF (where $f(x) \neq 0$).

Correct: If $f(x)$ is defined on $[0, 1]$, then $E(X) = \int_0^1 x \cdot f(x) \, dx$.

❌ Mistake: Confusing $E(X^2)$ with $[E(X)]^2$

Wrong: Using $\text{Var}(X) = E(X^2) - E(X)^2$ but calculating $E(X^2)$ as $(E(X))^2$.

Why it's wrong: $E(X^2) = \int x^2 f(x) \, dx$ is NOT the same as squaring $E(X)$!

Correct: Calculate $E(X^2)$ and $E(X)$ separately, then use $\text{Var}(X) = E(X^2) - [E(X)]^2$.