Worksheet: Variance of Discrete Random Variables

In this worksheet, we will practice calculating the variance of discrete random variables.

Q1:

Suppose 𝑋 and 𝑌 are independent, Var(𝑋)=24, and Var(𝑌)=30. Determine Var(7𝑋+9𝑌).

Q2:

Suppose 𝑋 and 𝑌 are independent, Var(𝑋)=4, and Var(𝑌)=18. Determine Var(3𝑋+5𝑌).

Q3:

Suppose 𝑋 and 𝑌 are independent, Var(𝑋)=9, and Var(𝑌)=36. Determine Var(5𝑋+8𝑌).

Q4:

Suppose that 𝑋, 𝑌 are two independent variables. Given that 𝐸(𝑌)=3, 𝐸(𝑋𝑌)=9, and 𝐸𝑋=30, find Var(𝑋).

  • A3
  • B6
  • C9
  • D21
  • E30

Q5:

If 𝑋 and 𝑌 are two independent random variables, is 𝐸(𝑋𝑌)=𝐸(𝑋)𝐸(𝑌)?

  • ANo
  • BYes

Is VarVarVar(𝑋𝑌)=(𝑋)(𝑌)?

  • ANo
  • BYes

Q6:

Suppose 𝑋 and 𝑌 are two independent variables such that Var(𝑋+𝑌)=2, where 𝑋 is a constant. Determine Var(𝑌).

Q7:

Suppose 𝑋 and 𝑌 are two independent variables such that Var(𝑋𝑌)=12 and Var(𝑌)=1. Determine Var(𝑋).

Q8:

Suppose 𝑋 and 𝑌 are two independent variables such that Var(𝑋)=2 and Var(𝑌)=5. Determine Var(𝑋2𝑌).

Q9:

Suppose that 𝑋, 𝑌 are two independent variables. Given that Var(𝑌)=1, 𝐸(𝑋)=2, and 𝐸𝑌=5, find 𝐸(𝑋𝑌).

Q10:

Suppose that 𝑋 and 𝑌 are two independent variables.

If Var(𝑋2𝑌)=9 and Var(𝑌)=2, determine Var(𝑋).

If 𝐸𝑋=5, find 𝐸(𝑋).

If 𝐸𝑌=6, find 𝐸(𝑌).

Find 𝐸(𝑋𝑌).

Q11:

Suppose 𝑋 and 𝑌 are two independent variables such that Var(𝑋+𝑌)=4. If 𝑋 is a constant, determine Var(𝑌).

Q12:

Let 𝑋 and 𝑌 be two independent random Variables.

Suppose that we know Var(2𝑋+𝑌)=6 and Var(𝑋+2𝑌)=5.

Find Var(𝑋) and Var(𝑌).

  • AVar(𝑋)=6 and Var(𝑌)=5
  • BVar(𝑋)=1915 and Var(𝑌)=1415
  • CVar(𝑋)=5 and Var(𝑌)=6
  • DVar(𝑋)=1915 and Var(𝑌)=1417
  • EVar(𝑋)=2917 and Var(𝑌)=1417

Q13:

The function in the given table is a probability function of a discrete random variable 𝑋. Find the variance of 𝑋. If necessary, give your answer to two decimal places.

𝑥3578
𝑓(𝑥)2𝐴5𝐴5𝐴𝐴

Q14:

The function in the given table is a probability function of a discrete random variable 𝑋. Given that the expected value of 𝑋 is 92, find the variance of 𝑋. Give your answer to two decimal places.

𝑥12𝐵7
𝑓(𝑥)10𝑎4𝑎126𝑎

Q15:

Let 𝑋 denote a discrete random variable which can take the values 2,1,𝑀,2and. Given that 𝑋 has probability distribution function 𝑓(𝑥)=𝑥+416, find the variance of 𝑋.

  • A0
  • B1516
  • C13564
  • D6516

Q16:

Let 𝑋 denote a discrete random variable which can take the values 3, 4, and 5. Given that 𝑓(𝑥)=𝑎𝑥12, find the variance of 𝑋. If necessary, give your answer to two decimal places.

Q17:

Let 𝑋 denote a discrete random variable which can take the values 2, 3, 5, and 8. Given that 𝑃(𝑋=2)=124, 𝑃(𝑋=3)=512, 𝑃(𝑋=5)=38, and 𝑃(𝑋=8)=16, find the variance of 𝑋. Give your answer to two decimal places.

Q18:

Given that Var(𝑋)=28, find Var(4𝑋+9).

Q19:

Let 𝑋 denote a discrete random variable which can take the values 2,4,5and. Given that 𝑋 has probability distribution function 𝑓(𝑥)=𝑎7𝑥+7, find the variance of 𝑋. Give your answer to two decimal places.

Q20:

Let 𝑋 denote a discrete random variable which can take the values 0,2,5and. Given that 𝑋 has probability distribution function 𝑓(𝑥)=𝑎6𝑥+6, find the variance of 𝑋.

  • A8
  • B83
  • C53
  • D31

Q21:

Let 𝑋 denote a discrete random variable which can take the values 1, 2, 7, and 8. Given that 𝑃(𝑋=1)=83, 𝑃(𝑋=2)=49, and 𝑃(𝑋=7)=118, find the variance of 𝑋. Give your answer to two decimal places.

Q22:

Given that Var(𝑋)=9, find Var(8𝑋+𝑋).

Q23:

Let 𝑋 denote a discrete random variable which can take the values 1,𝑎,7and. Given that 𝑋 has probability distribution function 𝑓(𝑥)=𝑥+218, find the variance of 𝑋. Give your answer to two decimal places.

Q24:

An experiment that produces the discrete random variable 𝑋 has the probability distribution shown.

𝑥2345
𝑝(𝑥)0.10.30.20.4

Calculate 𝐸(𝑋).

Calculate 𝐸𝑋.

The variance of 𝑋 can be calculated using the formula Var(𝑋)=𝐸𝑋𝐸(𝑋). Calculate Var(𝑋) to 2 decimal places.

Q25:

The probability mass functions for 𝑋 and 𝑌 are shown. Calculate the variance of 𝑋 and then the variance of 𝑌. If necessary, round your answer to the nearest hundredth.

𝑋872
𝑃(𝑋)0.170.330.5
𝑌9547
𝑃(𝑌)0.30.40.240.06
  • Avariance of 𝑋 is 2.69, variance of 𝑌 is 2.03
  • Bvariance of 𝑋 is 4.11, variance of 𝑌 is 7.24
  • Cvariance of 𝑋 is 2.03, variance of 𝑌 is 2.69
  • Dvariance of 𝑋 is 7.24, variance of 𝑌 is 4.11
  • Evariance of 𝑋 is 2.69, variance of 𝑌 is 4.11

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.