Continuous random variables

Welcome to the World of Continuous Random Variables!

In your H2 Mathematics journey, you’ve likely worked with discrete random variables—things you can count, like the number of heads in a coin toss. In Further Mathematics (9649), we move into the "smooth" world of Continuous Random Variables (CRVs). Instead of counting, we are measuring. Think about the exact time you wait for a bus, or the precise height of a tree; these values can be anything within a range, not just whole numbers!

By the end of these notes, you’ll understand how to use calculus to find probabilities, averages, and spread in this continuous world. Let’s dive in!

1. The Probability Density Function (PDF)

For a discrete variable, we use a probability mass function. For a CRV, we use a Probability Density Function (PDF), usually written as \( f(x) \).

Crucial Point: In the continuous world, the probability of the variable being exactly one value is zero, i.e., \( P(X = c) = 0 \). Instead, we find the probability of \( X \) falling within a range. This is represented by the area under the curve.

Two Golden Rules for a PDF:

1. The function can never be negative: \( f(x) \ge 0 \) for all \( x \).
2. The total area under the entire curve must be 1: \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \).

Piecewise Functions

Sometimes, a PDF is "split" into different parts. This is called a piecewise function. For example:
\( f(x) = \begin{cases} kx & 0 \le x \le 2 \\ 0 & \text{otherwise} \end{cases} \)
To find \( k \), you would integrate \( kx \) from 0 to 2 and set the result to 1!

Quick Review: To find the probability \( P(a \le X \le b) \), just calculate the definite integral: \( \int_{a}^{b} f(x) \, dx \).

2. Mean, Variance, and Expected Values

Just like in discrete statistics, we want to know the "average" and the "spread" of our data. Since we are dealing with continuous functions, we use integration instead of summation.

The Mean (Expectation)

The mean, \( E(X) \) or \( \mu \), is the "balance point" of the distribution.
\( E(X) = \int_{-\infty}^{\infty} x f(x) \, dx \)

Variance and Standard Deviation

Variance measures how spread out the values are from the mean.
Formula: \( Var(X) = E(X^2) - [E(X)]^2 \)
To find \( E(X^2) \), use: \( \int_{-\infty}^{\infty} x^2 f(x) \, dx \).

Expectation of a Function: \( E(g(X)) \)

If you need to find the expected value of a function of \( X \), such as \( X^3 \) or \( \sin(X) \), use this rule:
\( E(g(X)) = \int_{-\infty}^{\infty} g(x) f(x) \, dx \)

Common Mistake to Avoid: A very common error is thinking that \( E(X^2) \) is the same as \( (E(X))^2 \). They are very different! Always calculate the integral of \( x^2 f(x) \) separately.

3. Mode and Median

In addition to the mean, we can find other "centers" for our data.

The Mode

The Mode is the value of \( x \) where the PDF \( f(x) \) is at its maximum.
How to find it: Look for the highest point on the graph. If the function is differentiable, you can find it by setting \( f'(x) = 0 \) and checking if it's a maximum.

The Median

The Median (let's call it \( m \)) is the value that splits the area exactly in half (0.5 on the left, 0.5 on the right).
How to find it: Solve for \( m \) in the equation: \( \int_{-\infty}^{m} f(x) \, dx = 0.5 \).

Key Takeaway: The Mean is the "arithmetic average," the Mode is the "most frequent," and the Median is the "middle value." In skewed distributions, these three will all be different!

4. The Cumulative Distribution Function (CDF)

The Cumulative Distribution Function, written as \( F(x) \), tells us the probability that \( X \) is less than or equal to a certain value \( x \).

\( F(x) = P(X \le x) = \int_{-\infty}^{x} f(t) \, dt \)

The Link Between PDF and CDF

Think of the PDF as the gradient and the CDF as the area. This gives us a beautiful calculus relationship:
1. To get CDF from PDF: Integrate the PDF.
2. To get PDF from CDF: Differentiate the CDF: \( f(x) = \frac{d}{dx} F(x) \).

Did you know? The CDF always starts at 0 (on the far left) and ends at 1 (on the far right) because the total probability must accumulate to 1.

5. Special Probability Models

The syllabus identifies two specific continuous models you must master.

A. The Uniform (Rectangular) Distribution

This is the simplest model. The probability is constant over a specific interval \( [a, b] \). The graph looks like a rectangle!
PDF: \( f(x) = \frac{1}{b-a} \) for \( a \le x \le b \).
Mean: \( E(X) = \frac{a+b}{2} \) (The exact middle).
Variance: \( Var(X) = \frac{(b-a)^2}{12} \).

B. The Exponential Distribution

This model is often used to represent the time between events (like the time between customers entering a shop). It depends on a rate parameter, \( \lambda \).
PDF: \( f(x) = \lambda e^{-\lambda x} \) for \( x \ge 0 \).
Mean: \( E(X) = \frac{1}{\lambda} \).
Variance: \( Var(X) = \frac{1}{\lambda^2} \).
CDF: \( F(x) = 1 - e^{-\lambda x} \).

Analogy: Imagine you are waiting for a shooting star. If they occur randomly at a constant average rate, the time you wait follows an Exponential Distribution.

6. The Poisson and Exponential Connection

This is a favorite topic for exam questions! There is a direct "marriage" between the discrete Poisson distribution and the continuous Exponential distribution.

If the number of events occurring in a fixed interval follows a Poisson distribution with mean \( \lambda \)...
...then the time interval between those same events follows an Exponential distribution with the same parameter \( \lambda \).

Memory Aid:
Poisson = People (counting how many).
Exponential = Elapsed time (measuring how long).

Summary Checklist for Students

Before you tackle practice papers, make sure you can:
• Show that a function is a valid PDF (integral = 1).
• Calculate \( P(a < X < b) \) using integration.
• Switch between PDF \( f(x) \) and CDF \( F(x) \) using calculus.
• Find the Mean, Variance, Mode, and Median for any given PDF.
• Identify and use the formulas for Uniform and Exponential distributions.
• Explain how the Poisson and Exponential distributions relate to each other.

Don't worry if the integration seems tricky at first! Focus on setting up the correct boundaries for your integrals, and the rest is just applying the calculus rules you already know.