Worksheet: Standard Deviation and Variance of Discrete Random Variables

In this worksheet, we will practice calculating the variance and standard deviation of a probability distribution.

Q1:

The function in the given table is a probability function of a discrete random variable 𝑋. Find the standard deviation of 𝑋. Give your answer to two decimal places.

𝑥5431
𝑓(𝑥)131814724

Q2:

The function in the given table is a probability function of a discrete random variable 𝑋. Find the standard deviation of 𝑋. Give your answer to two decimal places.

𝑥4610
𝑓(𝑥)819819319

Q3:

The function in the given table is a probability function of a discrete random variable 𝑋. Given that the expected value of 𝑋 is 6.5, find the standard deviation of 𝑋. Give your answer to two decimal places.

𝑥3𝐴68
𝑓(𝑥)0.20.10.10.6

Q4:

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

Q5:

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

Q6:

Can the function in the given table be a probability distribution function?

𝑥0145
𝑓(𝑥)0.170.430.690.36
  • Ano
  • Byes

Q7:

A die is rolled two successive times. If the random variable 𝑋 denotes the sum of numbers appearing on the upper face after each roll, find the range of 𝑋.

  • AThe range of the random variable 𝑋={2,3,4,5,6,7,8,9,10,11,12}
  • BThe range of the random variable 𝑋={3,4,5,6,7,8,9,10,11}
  • CThe range of the random variable 𝑋={2,3,4,5,6,7,8,9,10,11}
  • DThe range of the random variable 𝑋={1,2,3,4,5,6}
  • EThe range of the random variable 𝑋={3,4,5,6,7,8,9,10,11,12}

Q8:

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 standard deviation of 𝑋. Give your answer to 2 decimal places.

Q9:

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

Q10:

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

Q11:

Given that Var(𝑋)=28, find Var(4𝑋+9).

Q12:

Let 𝑋 denote a discrete random variable which can take the values 2,2,4 and 5. Given that 𝑃(𝑋=2)=0.15, 𝑃(𝑋=2)=0.43, and 𝑃(𝑋=4)=0.25, find the value of 𝑃(𝑋>2).

Q13:

Let 𝑋 denote a discrete random variable which can take the values 3, 5, and 6. Which of the following functions could represent the probability function of 𝑋?

  • A𝑓(𝑥)=𝑥+310
  • B𝑓(𝑥)=𝑥28
  • C𝑓(𝑥)=45𝑥+2
  • D𝑓(𝑥)=4𝑥+52

Q14:

Which of the following is not a discrete random variable?

  • AThe number of cars in a family.
  • BThe number of children in a family.
  • CThe average height of children in a family.
  • DThe number of children taller than 4 ft in a family.

Q15:

Let 𝑋 denote a discrete random variable which can take the values 2, 6, 7, and 8. Given that 𝑃(𝑋=2)=𝑃(𝑋=6)=322 and 𝑃(𝑋=7)=411, find 𝑃(𝑋=8). Give your answer as a fraction.

  • A711
  • B311
  • C411
  • D111

Q16:

Let 𝑋 denote a discrete random variable which can take the values 1,2,3,4,5and. Given that 𝑋 has probability distribution function 𝑓(𝑥)=𝑎+𝑥16, find the value of 𝑎.

  • A15
  • B175
  • C3
  • D315

Q17:

The function in the given table is a probability function of a discrete random variable 𝑋. Find the value of 𝑎.

𝑥0123
𝑓(𝑥)3𝑎8𝑎4𝑎8𝑎
  • A1
  • B111
  • C1
  • D112

Q18:

Let 𝑋 denote a discrete random variable which can take the values 1,2,3 and 4. Given that 𝑃(𝑋=1)=0.46,𝑃(𝑋=2)=0.18, and 𝑃(𝑋=3)=0.14, find the value of 𝑃(𝑋=4).

Q19:

Let 𝑋 denote a discrete random variable which can take the values 0, 1, 3, and 4. Given that 𝑃(𝑥=0)=𝑚10, 𝑃(𝑥=1)=𝑚410, 𝑃(𝑥=3)=9𝑚+210, and 𝑃(𝑥=4)=10𝑚+310, find the value of 𝑚.

  • A121
  • B1121
  • C811
  • D37

Q20:

Let 𝑋 denote a discrete random variable which can take the values 1, 2, 3, 4, 5, and 6. Given that 𝑋 has probability distribution function 𝑓(𝑥)=𝑎𝑥5, find the value of 𝑎.

  • A1021
  • B57
  • C215
  • D521

Q21:

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.

Q22:

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.

Q23:

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.

Q24:

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

Q25:

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.

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