FAD1015 Week 6 — Continuous Random Variables

Week 6 lecture covering the cumulative distribution function (CDF), expectation, and variance for continuous random variables. The lecture consists of two parts: LEC 11 focuses on the CDF and its properties, while LEC 12 develops mean and variance for continuous cases. Source file: FAD1015 Week 6 cont-mean-var.pdf (28 slides).

Summary

This lecture extends random variable concepts to the continuous case. LEC 11 defines the cumulative distribution function $F(x)$, establishes its key properties, and demonstrates how to move between the probability density function $f(x)$ and the CDF via integration and differentiation. LEC 12 defines the expected value $E(X)$ and variance $\text{Var}(X)$ for continuous random variables, develops the expectation of a function $E[g(X)]$, and states the fundamental rules for linear transformations of random variables.

Key Concepts

  • Probability Distributions — Continuous random variables (general theory)
  • Cumulative Distribution Function (CDF)
  • Probability Density Function (PDF)
  • Expected Value / Mean of a Continuous Random Variable
  • Variance and Standard Deviation of a Continuous Random Variable
  • Expectation of a Function $g(X)$
  • Rules for $E[aX + b]$ and $\text{Var}(aX + b)$

LEC 11: Cumulative Distribution Function

Definition

If $X$ is a continuous random variable with probability density function $f(x)$, the cumulative distribution function $F(t)$ is defined as:

$$F(t) = P(X \leq t) = \int_{-\infty}^{t} f(x),dx$$

Properties of the CDF

  1. Boundary values: If the domain of $X$ is $x_0 \leq x \leq x_1$, then $F(x_0) = 0$ and $F(x_1) = 1$.

  2. Interval probability: For any $a, b$ in the domain, $$P(a \leq X \leq b) = P(a \leq X < b) = P(a < X \leq b) = P(a < X < b) = \int_a^b f(x),dx = F(b) - F(a)$$ (All inequalities are equivalent because $P(X = x) = 0$ for a continuous random variable.)

  3. Median: The median $m$ bisects the distribution such that $X$ is equally likely to be smaller or larger than $m$: $$F(m) = \int_{x_0}^{m} f(x),dx = \int_{m}^{x_1} f(x),dx = \frac{1}{2}$$

  4. PDF from CDF: Differentiating the CDF recovers the PDF: $$\frac{d}{dx}F(x) = f(x)$$

Worked Examples

The lecture presents six examples progressing in complexity:

  • Example 1: $f(x) = \dfrac{6 - 2x}{9}$ for $0 \leq x \leq 3$. Find $F(x)$, $P(X \leq 1.3)$, $P(1 \leq X \leq 2)$, and the median.
  • Example 2: $f(x) = ax - 3x^2$ for $0 \leq x \leq 2$. Find the constant $a$ and $F(x)$.
  • Example 3: $f(x) = \dfrac{3}{8}(1 + x^2)$ for $-1 \leq x \leq 1$. Find $F(x)$ and the median.
  • Example 4: Piecewise PDF: $f(x) = \dfrac{x}{25}$ ($0 \leq x < 5$) and $f(x) = \dfrac{2}{5} - \dfrac{x}{25}$ ($5 \leq x \leq 10$). Sketch, find $F(x)$, and compute several interval probabilities.
  • Example 5: CDF given as $F(x) = \dfrac{x^2}{k}$ ($0 \leq x < 2$). Find $k$, the median, and $f(x)$.
  • Example 6: Piecewise CDF: $F(x) = \dfrac{x^2}{48}$ ($0 \leq x < 4$) and $F(x) = -\dfrac{1}{2} + \dfrac{x}{4} - \dfrac{x^2}{96}$ ($4 \leq x < 12$). Compute probabilities, find $f(x)$, sketch $f(x)$ to find the mode, and find the median.

LEC 12: Mean and Variance for Continuous Random Variables

Expectation of $X$

For a continuous random variable $X$ with PDF $f(x)$, the mean (or expectation) is:

$$\mu = E(X) = \int_{\text{all }x} x,f(x),dx$$

Expectation of a Function $g(X)$

If $g(X)$ is a function of the random variable, then:

$$E[g(X)] = \int_{-\infty}^{\infty} g(x),f(x),dx$$

A frequently used case is $g(X) = X^2$:

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

Rules for the Expectation of $g(X)$

For constants $a$ and $b$:

  1. $E(a) = a$
  2. $E(aX) = a,E(X)$
  3. $E(aX + b) = a,E(X) + b$

Variance of $X$

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

Computational forms:

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

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

Standard deviation:

$$\sigma = \sqrt{\text{Var}(X)}$$

Rules for the Variance of $g(X)$

For constants $a$ and $b$:

  1. $\text{Var}(a) = 0$
  2. $\text{Var}(aX) = a^2,\text{Var}(X)$
  3. $\text{Var}(aX + b) = a^2,\text{Var}(X)$ (adding a constant does not affect spread)

Worked Examples

  • Example 7: $f(x) = \dfrac{1}{18}(6 - x)$ for $0 \leq x \leq 6$. Find $E(X)$.
  • Example 8: Car-park stay time with $f(x) = kx^{-3/2}$ for $1 \leq x \leq 9$. Interpret the domain, show $k = \dfrac{3}{4}$, and calculate the mean stay.
  • Example 9: Prove $E(aX + b) = aE(X) + b$ from the definition.
  • Example 10: $f(x) = 3(1 - x)^2$ for $0 \leq x \leq 1$. Find the mean, variance, standard deviation, $E(3X + 2)$, and $\text{Var}(3X + 2)$.
  • Example 11: Piecewise PDF: $f(x) = \dfrac{1}{4}$ ($0 \leq x < 2$) and $f(x) = \dfrac{x}{2} - \dfrac{3}{4}$ ($2 \leq x < 3$). Find $E(X)$, $\text{Var}(X)$, $\sigma$, $E(5X - 2)$, and $\text{Var}(5X - 2)$.
  • Example 12: Re-uses the CDF from Example 6. Find $E(X)$, $\text{Var}(X)$, $\sigma_X$, $E(Y)$ if $Y = 2X - 1$, $\text{Var}(Y)$ if $Y = 4X + 1$, and the standard deviation of $Y$, $\sigma_Y$.

Past Exam Questions (Tutorial / Revision)

The final slides list past-year examination titles only; the actual question text is not shown in the lecture slides:

  • Question 4 (2016/2017)
  • Question 5 (2016/2017)
  • Question 6.a (2016/2017)
  • Question 1.b (2015/2016)
  • Question 2 (2015/2016)

Related Topics

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