Discrete random variables (DRVs) and expectation

Welcome to Discrete Random Variables!

In this chapter of your AQA Further Mathematics course, we are diving into the world of Discrete Random Variables (DRVs) and Expectation. Essentially, we are looking at how to predict the "average" outcome of games, experiments, or real-world events where the results are numerical and countable.

Don't worry if this seems a bit abstract at first. Think of a DRV as a way of putting a number on luck! Whether you're a math whiz or find statistics a bit "hit or miss," these notes will break everything down into simple, manageable steps.

1. What is a Discrete Random Variable?

A Random Variable is just a quantity whose value depends on the outcome of a random event. We usually use a capital letter, like \( X \), to represent the variable itself, and a lowercase letter, like \( x \), to represent the actual values it can take.

Discrete means the variable can only take specific, separate values (like 1, 2, 3...) rather than any value in a range (like height or weight).
Example: The number of heads you get when flipping a coin three times is a DRV. You can get 0, 1, 2, or 3 heads, but you can't get 1.5 heads!

Probability Distributions

We usually show a DRV using a Probability Distribution Table. This table lists every possible value \( x \) and the probability of that value occurring, \( P(X=x) \).

The Golden Rule: The sum of all probabilities in a distribution must equal 1.
\( \sum P(X=x) = 1 \)

Sometimes, instead of a table, you'll be given a Probability Mass Function (PMF). This is just a formula that tells you how to calculate the probability for any given \( x \).
Example: \( P(X=x) = kx \) for \( x = 1, 2, 3 \). To find \( k \), you would solve \( k(1) + k(2) + k(3) = 1 \).

Quick Review Box:
• Discrete: Countable values (no decimals in between).
• Random Variable: A number based on a random outcome.
• Sum of Probabilities: Always 1!

2. Measures of Average and Spread

Just like with basic data, we want to know the "middle" and the "spread" of our DRV. These are the key measures you need to know:

The Mode: The value of \( x \) with the highest probability. It’s the "most likely" outcome.

The Median: The "middle" value. To find it, add up the probabilities from the smallest \( x \) until you reach or exceed 0.5. The value of \( x \) where this happens is your median.

The Mean (Expected Value): This is the average value you would expect to get if you ran the experiment many, many times. In Further Maths, we call this Expectation, written as \( E(X) \).

The Formula for Expectation

To find the mean, you multiply each value by its probability and add them all up:
\( E(X) = \sum x_i p_i \)

The Formula for Variance

Variance, written as \( Var(X) \), measures how spread out the values are from the mean. A high variance means the results are very unpredictable. To calculate it, we use this "MS-SM" trick:

The "MS-SM" Mnemonic:
Mean of the Squares minus the Square of the Mean.

1. Find \( E(X^2) \): Square each \( x \) value, multiply by its probability, and sum them up: \( E(X^2) = \sum x_i^2 p_i \).
2. Square the mean: \( (E(X))^2 \).
3. Subtract them: \( Var(X) = E(X^2) - (E(X))^2 \).

Standard Deviation: This is simply the square root of the variance: \( \sigma = \sqrt{Var(X)} \).

Key Takeaway: \( E(X) \) is the average result. \( Var(X) \) is how much the results "swing" away from that average.

3. Expectation of Functions of X

Sometimes we don't just want the mean of \( X \); we might want the mean of \( X^2 \), \( 5X^3 \), or even \( 18X^{-3} \). The rule is simple: replace the \( x \) in the summation formula with whatever function you are looking for.

The General Rule: \( E(g(X)) = \sum g(x_i) p_i \)

Example: If you want to find \( E(5X^3) \), you calculate \( \sum (5x_i^3 \times P(X=x_i)) \).

Linear Transformations

If you change your variable by multiplying by a constant \( a \) and adding a constant \( b \), you don't have to redo the whole table! Use these handy shortcuts:

For Expectation: \( E(aX + b) = aE(X) + b \)
(The mean moves and stretches exactly how you'd expect).

For Variance: \( Var(aX + b) = a^2 Var(X) \)
(Important! Adding \( b \) doesn't change the spread, so \( b \) disappears. Also, the scaling factor \( a \) is squared because variance is a "squared" measure).

Common Mistake to Avoid: Many students think \( E(X^2) \) is the same as \( (E(X))^2 \). They are NOT! This is exactly why variance exists—the difference between these two values tells us about the spread.

4. The Discrete Uniform Distribution

Imagine a perfectly fair six-sided die. Every outcome (1, 2, 3, 4, 5, 6) has the exact same probability (\( 1/6 \)). This is a Discrete Uniform Distribution.

A discrete uniform distribution is defined on the set \( \{1, 2, 3, ..., n\} \), where each of the \( n \) values has a probability of \( 1/n \).

Special Formulae for Uniform Distributions

If \( X \) is a discrete uniform distribution from 1 to \( n \), you can use these shortcuts (which you are required to know the proofs for!):

Mean: \( E(X) = \frac{n + 1}{2} \)
(This is just the average of the first and last number).

Variance: \( Var(X) = \frac{n^2 - 1}{12} \)

Did you know? These formulae only work if the values start at 1 and go up in 1s to \( n \). If your values are different (e.g., 10, 20, 30), you should use the linear transformation rules from the previous section to adapt the formula!

Proof of the Mean (SA7)

To prove \( E(X) = \frac{n+1}{2} \):
1. Use the definition: \( E(X) = \sum_{x=1}^n x \times \frac{1}{n} \).
2. Factor out the \( \frac{1}{n} \): \( E(X) = \frac{1}{n} \sum_{x=1}^n x \).
3. Use the sum of integers formula \( \sum x = \frac{n(n+1)}{2} \).
4. Multiply: \( E(X) = \frac{1}{n} \times \frac{n(n+1)}{2} = \frac{n+1}{2} \).

Key Takeaway Summary:
• For any DRV: \( E(X) = \sum xp \) and \( Var(X) = E(X^2) - [E(X)]^2 \).
• For Linear shifts: \( E(aX+b) = aE(X)+b \) and \( Var(aX+b) = a^2Var(X) \).
• For Uniform (1 to \( n \)): Mean = \( \frac{n+1}{2} \), Var = \( \frac{n^2-1}{12} \).

You've reached the end of the notes for this section! Great job. Keep practicing with different probability tables, and soon finding the "Expectation" will be second nature.