Probabilities with combined normal variables

10 minA Level6.2.1

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find probabilities about a total of several normal quantities, and decide which of two normal quantities is likely to be bigger by rewriting $P(X > Y)$ as $P(X - Y > 0)$ .

Key to the symbols

Symbol	Meaning
$X \sim N(\mu, \sigma^2)$	$X$ is normal with mean $\mu$ and variance $\sigma^2$
$T$	a total, such as $X_1 + X_2 + X_3$
$D$	a difference, such as $X - Y$
$Z$ , $\Phi(z)$	the standard normal variable and the area to the left of $z$

Definition

Combinations of normals are normal

If $X$ and $Y$ are independent and both normally distributed, then any combination $aX + bY$ is also normally distributed. The same holds for totals of several independent normal variables.

Its mean and variance come from the rules in the earlier lessons:

$aX + bY \sim N\big(aE(X) + bE(Y),\ a^2\text{Var}(X) + b^2\text{Var}(Y)\big)$

This matters because it lets you use the normal table on the combination.

The four-step method

Define the combination as one new variable, for example $T = X_1 + X_2 + X_3 + X_4$ or $D = X - Y$ .
Mean: add up the means (with the multipliers).
Variance: add up the variances (with the multipliers squared), then square-root to get the standard deviation.
Standardise and use the normal table.

Worked example: Example 1: will the lift be overloaded?

Adult masses are modelled by $N(75, 100)$ in kg. A lift has a safety limit of 320 kg. Four adults chosen independently get in. Find the probability that their total mass exceeds the limit.

Have a go first?

Check 1

CalculateQuestion 1 · Apply

Answer the question above to continue.