CAIE A-Level Mathematics 5.4.3 Expectation and Variance

Practise using standard expectation and variance formulae for binomial variables and the expected waiting time for geometric variables, with parameters read from context.

Exam points

for X distributed B(n, p), calculate E(X) = np and Var(X) = np(1 − p) from its parameters
for X distributed Geo(p), calculate the expected trial of first success as E(X) = 1/p
identify n and p from the experiment before substituting and interpret the resulting mean

[Maximum number: 2]

An ordinary fair die is thrown repeatedly until a 1 or a 6 is obtained.

On another occasion, this die is thrown 3 times. The random variable X is the number of times that a 1 or a 6 is obtained.

Find E(X).

$E(X)=0\left(\frac8{27}\right)+1\left(\frac{12}{27}\right)+2\left(\frac6{27}\right)+3\left(\frac1{27}\right).$
M1

$E(X)=\frac{12}{27}+\frac{12}{27}+\frac3{27}=1.$
A1

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5.4.3—Expectation and variance question 1