Exercises for Question Type 1: Finding the expected value or variance of a transformed variable given some information on its summary statistics

Exercises for Question Type 1: Finding the expected value or variance of a transformed variable given some information on its summary statistics - IB

Written by official IB examiners

Question 1

Skill question

Random variables $X$ and $Y$ are independent with standard deviations $σ_{X} = 3$ and $σ_{Y} = 2$ . Given $E (X + 3 Y) = 3$ , find $E (3 X + 9 Y)$ and $Var (3 X + 9 Y)$ .

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Question 2

Skill question

Given a random variable $X$ has $Var (X) = 2$ , calculate $Var (5 X + 3)$ .

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Question 3

Skill question

The question tests the properties of variance for linear combinations of independent random variables, specifically $Var (a X + bY + c) = a^{2} Var (X) + b^{2} Var (Y)$ for independent $X$ and $Y$ .

If $X$ and $Y$ are independent random variables with $Var (X) = 6$ and $Var (Y) = 2$ , find $Var (X - 2 Y + 5)$ .

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Question 4

Skill question

In a portfolio model, $X$ and $Y$ represent returns on two independent assets with $E (X) = 0.05$ , $E (Y) = 0.02$ , $σ_{X} = 0.1$ , and $σ_{Y} = 0.08$ . An investor's total return is $R = 0.6 X + 0.4 Y$ .

Find $E (R)$ and $Var (R)$ .

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Question 5

Skill question

Independent random variables $X$ , $Y$ , and $Z$ satisfy $E (X) = 2$ , $E (Y) = 1$ , $E (Z) = 3$ , $Var (X) = 1$ , $Var (Y) = 4$ , and $Var (Z) = 9$ .

Find $E (2 X - Y + 3 Z)$ and $Var (2 X - Y + 3 Z)$ .

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Question 6

Skill question

Let $X$ , $Y$ , and $Z$ be independent random variables with standard deviations $σ_{X} = 2$ , $σ_{Y} = 1$ , $σ_{Z} = 3$ and expectations $E (X) = 1$ , $E (Y) = 2$ , $E (Z) = 0$ .

Find $E (X + Y + Z)$ and $Var (X + Y + Z)$ .

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Question 7

Skill question

Random variables $X$ and $Y$ are independent with $E (X) = 1$ , $E (Y) = 2$ , $Var (X) = 4$ , and $Var (Y) = 9$ . Find $E (3 X + 4 Y)$ and $Var (3 X + 4 Y)$ .

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Question 8

Skill question

Random variables $X$ and $Y$ have $Var (X) = 5$ , $Var (Y) = 8$ , and $Cov (X, Y) = 2$ . Compute $Var (2 X - Y)$ .

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Question 9

Skill question

Given a random variable $X$ has $E (X) = 4$ , calculate $E (5 X + 3)$ .

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Question 10

Skill question

For random variables $X$ and $Y$ with $E (X) = 3$ and $E (Y) = 4$ , determine $E (2 X - 4 Y + 7)$ .

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Question 11

Skill question

Properties of expectation and variance for discrete random variables

Given $E (X) = 2$ and $Var (X) = 3$ , find $E (2 X - 7)$ and $Var (2 X - 7)$ .

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Question 12

Skill question

Random variables $X$ and $Y$ satisfy $σ_{X} = 3$ , $σ_{Y} = 4$ , and $ρ_{X Y} = 0.6$ . Calculate $Var (4 X + 5 Y)$ .

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IB Mathematics Applications & Interpretation Question Type 1: Finding the Expected Value or Variance of a Transformed Variable Given Some Information on Its Summary Statistics Exercises