Mathematical Methods — Question 5

QCAA 2022, Paper 2 · 1 mark

Q5 · 2022 · Technology-activeSimple familiar1 mark

QUESTION 5

The continuous random variable

X

has the probability density function

f(x)=\begin{cases}\dfrac{\cos x}{2},&-\dfrac\pi2\le x\le\dfrac\pi2,\\0,&\text{otherwise.}\end{cases}

The standard deviation of

X

(A)

0.467

(B)

0.684

(C)

1.211

(D)

1.467

ABCD

WORKED SOLUTION

1 mark

ANSWER

B —

0.684

Worked solution

E(X)=0

by symmetry. Evaluating

E(X^2)=\int_{-\pi/2}^{\pi/2}x^2(\cos x/2)\,dx

gives variance about

0.4674

, so the standard deviation is about

0.684

; B.

Selects option B.

[1 mark]

Worked explanation by QCE Vault; correct option verified against the QCAA answer key.