Expected value and standard deviation of a random variable

Warm-up

Three quick questions from earlier lessons.

Discover

A raffle ticket wins 10 dollars with probability 0.1 and nothing otherwise. What is the average win per ticket over many tickets?

1 dollar
5 dollars

Key idea

The mean and SD of a random variable.

E(X) = \mu_X = \sum x\,P(x)

, the long-run average.

\text{Var}(X) = \sum (x - \mu)^2 P(x)

and

\text{SD}(X) = \sqrt{\text{Var}(X)}

. For

X

equal to 1 with probability

p

and 0 otherwise:

\mu = p

,

\text{Var} = p(1 - p)

.

Worked example

$X$ : 0, 1, 2 with probabilities 0.3, 0.4, 0.3.

E(X) = 0 + 0.4 + 0.6 = 1

Σ x P(x).

\text{Var}(X) = 0.3 + 0 + 0.3 = 0.6

Σ (x − 1)² P(x).

\text{SD}(X) = \sqrt{0.6} \approx 0.77

Square root.

Your turn, guided

Your turn: $X$ = 0, 5 with probabilities 0.6, 0.4. $E(X)$ .

2

5 × 0.4.

Practice ★

X takes the values 0, 1, 2 with probabilities 51/100, 15/100 and 34/100. Find E(X).
X takes the values 0, 1, 2 with probabilities 56/100, 25/100 and 19/100. Find E(X).

A game costs 1 dollar. You win 9 dollars (net) with probability 1/7; otherwise you lose your dollar. Find your expected net gain, in dollars.

Watch out!

Practice ★★

X = 1 with probability 7/10 and X = 0 otherwise. Find the standard deviation of X.
X = 1 with probability 9/10 and X = 0 otherwise. Find the standard deviation of X.

Exit ticket

Three questions to check you have got it.

X takes the values 0, 1, 2 with probabilities 51/100, 15/100 and 34/100. Find E(X).
X takes the values 0, 1, 2 with probabilities 56/100, 25/100 and 19/100. Find E(X).
X takes the values 0, 1, 2 with probabilities 77/100, 10/100 and 13/100. Find E(X).

Summary card

Expected value and SD

$E(X) = \sum x P(x)$ ; $\text{SD} = \sqrt{\sum (x - \mu)^2 P(x)}$ .

Watch out: weight by probabilities; include losses.

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