IB Maths AI HL Ahl 4 14 Hl Random Variable Transformations Questions

Practise IB Mathematics HL 4.14 by applying random variable transformations methods to questions from the Question Bank.

Exam points

Apply E(aX+b)=aE(X)+b and Var(aX+b)=a²Var(X) to transformed random variables.
Calculate expected value and variance for linear combinations of independent random variables.
Use sample mean and unbiased sample variance as estimators and interpret the resulting uncertainty.

[Maximum number: 3]

Taylor is playing a computer game in which they shoot at spaceships and battleships. The number of spaceships they hit per minute can be modelled by a Poisson distribution with mean 4.2. The number of battleships they hit per minute can be modelled by a Poisson distribution with a mean of 2.3. Any single hit occurs independently of all others.

(a)

Every spaceship that is hit earns Taylor 3 points and every battleship 5 points. Let T be the total points earned in one minute.

Find

[ 3 ]

(i)

E(T).

[ 1 ]

$\mathrm{E}(T)=3 \times 4.2+5 \times 2.3=24.1$

(ii)

$\quad \operatorname{Var}(T)$ .

[ 2 ]

$\quad \operatorname{Var}(T)=3^{2} \times 4.2+5^{2} \times 2.3=95.3$

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IB Maths AI HL Ahl 4 14 Hl Random Variable Transformations Questions question 1

Question (a)

Question (i)

Question (ii)