Topic 4.9 Notes – Combining Random Variables

1. Linear Transformations of a Random Variable

A linear transformation has the form
$$Y = a + bX$$

• a shifts every value (adds/subtracts a constant).
• b rescales every value (multiplies/divides by a constant).

What Happens to the Distribution

For $Y = a + bX$:

$$\mu_Y = a + b\mu_X$$

$$\sigma_Y = |b|\sigma_X$$

Notice two patterns:

• Adding/subtracting (a) changes the mean only, not the spread.
• Multiplying/dividing (b) changes both the mean and the standard deviation.
• The shape stays the same as long as $b > 0$.
• If $b < 0$, the distribution is reflected (like flipping it across a vertical axis).

To see this in action, compare a normal distribution for $X$ with the transformed variable $Y = 10 + 2X$:

Effect of $Y = 10 + 2X$ on a normal distribution

The new curve is still bell-shaped, but its center shifts and its spread increases because every value was multiplied by 2 and then shifted up by 10.

Quick Example

Suppose test scores have:

• $\mu_X = 70$
• $\sigma_X = 8$

If a teacher adds 5 points to every score:

• New mean: $70 + 5 = 75$
• New SD: still $8$

If instead every score is multiplied by 1.1 (curve scaled):

• New mean: $1.1(70) = 77$
• New SD: $1.1(8) = 8.8$

Students often think adding 5 changes the SD. It doesn’t. Spread only changes when you multiply or divide.

2. Linear Combinations of Random Variables

Now we combine two random variables.

A linear combination looks like:

$$aX + bY$$

This includes:

• Sums $X + Y$
• Differences $X - Y$
• Weighted combos like $2X + 0.5Y$

We care about how the mean and variance behave.

3. Means of Sums and Differences

This part is simple and always works, even if the variables are not independent.

$$\mu_{aX + bY} = a\mu_X + b\mu_Y$$

Special cases

• If $S = X + Y$, then
  $\mu_S = \mu_X + \mu_Y$
• If $D = X - Y$, then
  $\mu_D = \mu_X - \mu_Y$

Independence is not required for means.

Example

Let:

• $X$ = time (minutes) to finish Part A, $\mu_X = 30$
• $Y$ = time (minutes) to finish Part B, $\mu_Y = 45$

If $T = X + Y$, then:

$$\mu_T = 30 + 45 = 75$$

In context:
The mean total time to complete both parts is 75 minutes.

Always name the variable and the situation. That wording matters on FRQs.

4. Variance and Standard Deviation of Sums and Differences

This is where independence matters.

Two variables are independent if knowing one gives no information about the distribution of the other.

If $X$ and $Y$ are independent:

$$\text{Var}(aX + bY) = a^2\sigma_X^2 + b^2\sigma_Y^2$$

Then take the square root for SD.

Special cases

If $S = X + Y$:

$$\sigma_S = \sqrt{\sigma_X^2 + \sigma_Y^2}$$

If $D = X - Y$:

$$\sigma_D = \sqrt{\sigma_X^2 + \sigma_Y^2}$$

Notice something important:
You add variances even for subtraction.

Example

Suppose:

• $\sigma_X = 4$
• $\sigma_Y = 3$

If independent:

$$\sigma_{X+Y} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$

Students mess this up by:

• Adding SDs directly (4 + 3 = 7 ❌)
• Subtracting variances for differences ❌
• Forgetting to square coefficients in $aX + bY$

If independence is not stated, you cannot assume variances add.