Lesson Worksheet: Standard Deviation of Discrete Random Variables Mathematics

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

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:

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.

Q5:

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.

𝑥12410
𝑓(𝑥)0.2𝐴0.30.3

Q6:

13 students took an exam; 8 students got 6 marks, 3 students got 10 marks, and 2 students got 2 marks. Given that 𝑥 denotes the number of marks received, find the coefficient of variation of 𝑥 as a percentage to the nearest hundredth.

Q7:

Let 𝑋 denote a discrete random variable which can take the values 1, 4, and 6. Given that 𝑓(𝑥)=𝑥+11𝑎, find the coefficient of variance of 𝑋. If necessary, give your answer to two decimal places.

Q8:

Let 𝑋 denote a discrete random variable which can take the values 1, 0, and 7. Given that the expected value of 𝑋 is 3.8 and 𝑃(𝑥=7)=1425, find the coefficient of variation, as a percentage, to two decimal places.

Q9:

What is the variance of an exponentially distributed random variable with parameter 𝜆?

  • A1𝜆
  • B1𝜆
  • C𝜆
  • D𝜆

Q10:

Let 𝑋 denote a discrete random variable that can take the values 1, 0, and 1. Given that 𝑋 has a probability distribution function 𝑓(𝑥)=𝑎3𝑥, find the coefficient of variation to the nearest percent.

Q11:

Let 𝑋 denote the discrete random variable that can take values 0, 2, 4, and 6. Given that 𝑃(𝑋=0)=17, 𝑃(𝑋=2)=27, and 𝑃(𝑋=4)=27, find the standard deviation of 𝑋, giving your answer to two decimal places.

Q12:

Work out the standard deviation of the random variable 𝑋 whose probability distribution is shown. Give your answer to two decimal places.

Q13:

The function in the given table is a probability function of a discrete random variable 𝑋. Find the coefficient of variation of 𝑋. Give your answer to the nearest percent.

𝑋12345
𝑓(𝑋)11011015310310

Q14:

Work out the coefficient of variation of the random variable 𝑋 whose probability distribution is shown. Give your answer to the nearest percent.

Q15:

Let 𝑋 denote a discrete random variable that can take the values 0, 𝑎, and 4. Given that 𝑋 has a probability distribution function 𝑓(𝑥)=𝑥+19, find the coefficient of variation to the nearest percent.

Q16:

Let 𝑋 denote a discrete random variable that can take the values 2, 3, 5, and 7. Given that 𝑃(𝑋=2)=112, 𝑃(𝑋=3)=14, 𝑃(𝑋=5)=13, and 𝑃(𝑋=7)=13, find the coefficient of variation to the nearest percent.

Q17:

You have two boxes. Box 1 contains 3 identical cards numbered from 1 to 3, and box 2 contains 4 identical cards numbered from 1 to 4. Consider the experiment of choosing two cards from each box and let 𝑋 be the random variable that represents the average of the four selected numbers.

Calculate the standard deviation of 𝑋. Round your answer to three decimal places.

Calculate the coefficient of variation of 𝑋. Round your answer to the nearest percent.

Q18:

For a discrete random variable 𝑋, if the variance of 𝑋 is 1.132 and the coefficient of variation is 11%, calculate the mean to the nearest two decimal places.

Q19:

Two fair dice are rolled; one of them is four sided and the other is six sided. Let 𝑋 be the random variable that represents the average of the numbers.

Calculate the standard deviation of 𝑋. Round your answer to three decimal places.

Calculate the coefficient of variation of 𝑋. Round your answer to the nearest percent.

Q20:

Which of the following formulae is used to calculate the standard deviation of discrete random variables?

  • A𝜎=(𝑥𝜇)𝑃(𝑋=𝑥)
  • B𝜎=(𝑥𝜇)
  • C𝜎=(𝑥𝜇)𝑃(𝑋=𝑥)
  • D𝜎=((𝑥𝜇)𝑃(𝑋=𝑥))
  • E𝜎=((𝑥𝜇)𝑃(𝑋=𝑥))

Q21:

A pair of fair dice are thrown. Let 𝑋 be the random variable that represents the absolute difference of the two numbers on the dice.

Calculate the standard deviation of 𝑋. Round your answer to three decimal places.

Calculate the coefficient of variation of 𝑋. Round your answer to the nearest percent.

Q22:

You have two boxes, each containing three identical cards numbered from 1 to 3. A coin is tossed. If it falls heads up, you will select one card from each box. If it falls tails up, you will select one card from the first box and two cards from the second.

Let 𝑋 be the random variable that represents the sum of the numbers that appear on the selected cards.

Calculate the standard deviation of 𝑋. Round your answer to three decimal places.

Calculate the coefficient of variation of 𝑋. Round your answer to the nearest percent.

Q23:

Given that 𝑓(𝑥)=115𝑥, where 𝑥{1,2,3,,𝑘}, is a probability distribution function of a discrete random variable, calculate the standard deviation of 𝑋. Round your answer to three decimal places.

Q24:

The function in the given table is a probability distribution function of a discrete random variable.

𝑥0123
𝑓(𝑥)0.250.30.2𝑎

Find the standard deviation of 𝑋. Give your answer to two decimal places.

Q25:

Given that 𝑓(𝑥)=𝑎𝑥+1𝑘+20 is the probability distribution function of a discrete random variable that takes the values 0, 1, and 2. Determine the set of all ordered pairs (𝑎,𝑘) if the standard deviation is 0.51. Approximate the values of 𝑎 and 𝑘 to the nearest two decimal places.

  • A{(21.96,48.87)}
  • B{(21.96,48.87),(0.49,18.47)}
  • C{(1.46,12.62),(0.37,18.12)}
  • D{(0.29,16.13),(0.29,17.87)}
  • E{(1.46,12.62)}

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