Combining random variables

About this lesson

Statistics (AP-aligned) › Random variables

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Combining random variables

Warm-up

Three quick questions from earlier lessons.

Discover

Two independent dice each have SD about 1.7. Is the SD of their sum about 3.4?

No, smaller: about 2.4
Yes

Key idea

Means always add; variances add for independent variables.

E(a + bX) = a + bE(X)

,

\text{SD}(a + bX) = |b|\,\text{SD}(X)

.

E(X \pm Y) = E(X) \pm E(Y)

. If

X

and

Y

are independent,

\text{Var}(X \pm Y) = \text{Var}(X) + \text{Var}(Y)

: variances add even for a difference.

Worked example

Morning commute: mean 20, SD 3; evening: mean 25, SD 4; independent.

E(\text{total}) = 45

Means add.

\text{Var} = 9 + 16 = 25

Variances add.

\text{SD}(\text{total}) = 5

Square root.

Your turn, guided

Your turn: SD(X) = 6, SD(Y) = 8, independent. SD(X − Y).

10

√(36 + 64).

Practice ★

X and Y are independent with SD(X) = 27 and SD(Y) = 36. Find SD(X + Y).
X and Y are independent with SD(X) = 18 and SD(Y) = 24. Find SD(X + Y).

SD(X) = 8. Let Y = 12 + 5X. Find SD(Y).

Watch out!

Practice ★★

X and Y are independent with SD(X) = 24 and SD(Y) = 10. Find SD(X − Y).
X and Y are independent with SD(X) = 72 and SD(Y) = 30. Find SD(X − Y).

Exit ticket

Three questions to check you have got it.

X and Y are independent with SD(X) = 27 and SD(Y) = 36. Find SD(X + Y).
X and Y are independent with SD(X) = 18 and SD(Y) = 24. Find SD(X + Y).
X and Y are independent with SD(X) = 12 and SD(Y) = 16. Find SD(X + Y).

Summary card

Combining random variables

Means add (and subtract).
Independent: variances add, for sums and differences.

Watch out: never add SDs directly.

End of the lesson

Next lesson: The binomial distribution Back to the chapter

Answers: Combining random variables

A

$45$
$30$

B

$40$

C

$26$
$78$

D

$45$
$30$
$20$

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