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.

𝑥 5 4 3 1
𝑓 ( 𝑥 ) 1 3 1 8 1 4 7 2 4

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.

𝑥 4 6 10
𝑓 ( 𝑥 ) 8 1 9 8 1 9 3 1 9

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.

𝑥 3 𝐴 6 8
𝑓 ( 𝑥 ) 0.2 0.1 0.1 0.6

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.

𝑥 3 5 7 8
𝑓 ( 𝑥 ) 2 𝐴 5 𝐴 5 𝐴 𝐴

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 9 2 , find the variance of 𝑋 . Give your answer to two decimal places.

𝑥 1 2 𝐵 7
𝑓 ( 𝑥 ) 1 0 𝑎 4 𝑎 1 2 6 𝑎

Q6:

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

𝑥 0 1 4 5
𝑓 ( 𝑥 ) 0.17 0.43 0.69 0.36
  • 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 .

  • A The range of the random variable 𝑋 = { 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 }
  • B The range of the random variable 𝑋 = { 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 }
  • C The range of the random variable 𝑋 = { 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 }
  • D The range of the random variable 𝑋 = { 1 , 2 , 3 , 4 , 5 , 6 }
  • E The range of the random variable 𝑋 = { 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 }

Q8:

Let 𝑋 denote a discrete random variable which can take the values 0 , 2 , 5 a n d . Given that 𝑋 has probability distribution function 𝑓 ( 𝑥 ) = 𝑎 6 𝑥 + 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 , 2 a n d . Given that 𝑋 has probability distribution function 𝑓 ( 𝑥 ) = 𝑥 + 4 1 6 , find the variance of 𝑋 .

  • A0
  • B 1 3 5 6 4
  • C 6 5 1 6
  • D 1 5 1 6

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 ) = 𝑎 7 𝑥 + 7 , find the variance of 𝑋 . Give your answer to two decimal places.

Q11:

Given that V a r ( 𝑋 ) = 2 8 , find V a r ( 4 𝑋 + 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 . 1 5 , 𝑃 ( 𝑋 = 2 ) = 0 . 4 3 , and 𝑃 ( 𝑋 = 4 ) = 0 . 2 5 , 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 ?

  • A 𝑓 ( 𝑥 ) = 4 5 𝑥 + 2
  • B 𝑓 ( 𝑥 ) = 4 𝑥 + 5 2
  • C 𝑓 ( 𝑥 ) = 𝑥 + 3 1 0
  • D 𝑓 ( 𝑥 ) = 𝑥 2 8

Q14:

Which of the following is not a discrete random variable?

  • AThe number of children in a family.
  • BThe average height of children in a family.
  • CThe number of cars 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 ) = 3 2 2 and 𝑃 ( 𝑋 = 7 ) = 4 1 1 , find 𝑃 ( 𝑋 = 8 ) . Give your answer as a fraction.

  • A 4 1 1
  • B 3 1 1
  • C 7 1 1
  • D 1 1 1

Q16:

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

  • A 3
  • B 3 1 5
  • C 1 5
  • D 1 7 5

Q17:

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

𝑥 0 1 2 3
𝑓 ( 𝑥 ) 3 𝑎 8 𝑎 4 𝑎 8 𝑎
  • A1
  • B 1 1 2
  • C 1 1 1
  • D 1

Q18:

Let 𝑋 denote a discrete random variable which can take the values 1 , 2 , 3 and 4. Given that 𝑃 ( 𝑋 = 1 ) = 0 . 4 6 , 𝑃 ( 𝑋 = 2 ) = 0 . 1 8 , and 𝑃 ( 𝑋 = 3 ) = 0 . 1 4 , 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 ) = 𝑚 1 0 , 𝑃 ( 𝑥 = 1 ) = 𝑚 4 1 0 , 𝑃 ( 𝑥 = 3 ) = 9 𝑚 + 2 1 0 , and 𝑃 ( 𝑥 = 4 ) = 1 0 𝑚 + 3 1 0 , find the value of 𝑚 .

  • A 3 7
  • B 8 1 1
  • C 1 2 1
  • D 1 1 2 1

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 ) = 𝑎 𝑥 5 , find the value of 𝑎 .

  • A 5 7
  • B 1 0 2 1
  • C 2 1 5
  • D 5 2 1

Q21:

Let 𝑋 denote a discrete random variable which can take the values 1 , 𝑎 , 7 a n d . Given that 𝑋 has probability distribution function 𝑓 ( 𝑥 ) = 𝑥 + 2 1 8 , 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 ) = 𝑎 𝑥 1 2 , 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 ) = 1 2 4 , 𝑃 ( 𝑋 = 3 ) = 5 1 2 , 𝑃 ( 𝑋 = 5 ) = 3 8 , and 𝑃 ( 𝑋 = 8 ) = 1 6 , 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 , 5 a n d . Given that 𝑋 has probability distribution function 𝑓 ( 𝑥 ) = 𝑎 6 𝑥 + 6 , find the variance of 𝑋 .

  • A 5 3
  • B 8 3
  • C8
  • D31

Q25:

Let 𝑋 denote a discrete random variable which can take the values 1, 2, 7, and 8. Given that 𝑃 ( 𝑋 = 1 ) = 8 3 , 𝑃 ( 𝑋 = 2 ) = 4 9 , and 𝑃 ( 𝑋 = 7 ) = 1 1 8 , find the variance of 𝑋 . Give your answer to two decimal places.

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