H2 Maths 6.2 - Linear Transformation: E(aX + b) and Var(aX + b)

Step 1 - State the problem

X follows a binomial distribution with n equals eight and p equals zero point four.

Step 1 - State the problem

Each success triggers a two hundred dollar bonus. There is also a fixed one hundred and fifty dollar stipend.

Step 1 - State the problem

The total payout is T equals two hundred X plus one hundred and fifty.

Step 1 - State the problem

Find E of T and the variance of T.

Step 2 - Find E(X) and Var(X)

First, we need E of X and variance of X from the binomial formulas.

Step 2 - Find E(X) and Var(X)

E of X equals n p, which is eight times zero point four, giving three point two.

Step 2 - Find E(X) and Var(X)

Variance of X equals n p times one minus p.

Step 2 - Find E(X) and Var(X)

That is eight times zero point four times zero point six, which equals one point nine two.

Step 3 - Apply the linear transformation rules

For Y equals a X plus b, E of Y equals a times E of X plus b.

Step 3 - Apply the linear transformation rules

Here a is two hundred and b is one hundred and fifty.

Step 3 - Apply the linear transformation rules

E of T equals two hundred times three point two plus one hundred and fifty.

Step 3 - Apply the linear transformation rules

That is six hundred and forty plus one hundred and fifty, giving seven hundred and ninety dollars.

Step 3 - Apply the linear transformation rules

For the variance, the constant b drops out. Variance of Y equals a squared times variance of X.

Step 3 - Apply the linear transformation rules

That is two hundred squared times one point nine two.

Step 3 - Apply the linear transformation rules

Forty thousand times one point nine two equals seventy-six thousand eight hundred dollars squared.

Step 4 - Find standard deviation and interpret

The standard deviation is the square root of seventy-six thousand eight hundred.

Step 4 - Find standard deviation and interpret

That gives approximately two hundred and seventy-seven dollars.

Step 4 - Find standard deviation and interpret

So on average the payout is seven hundred and ninety dollars, and it typically varies by about two hundred and seventy-seven dollars from this mean.

Step 4 - Find standard deviation and interpret

Key takeaway: the constant one hundred and fifty shifts the mean but does not affect the spread.