Expected Value Combining Random Variables

At a large regional collegiate women’s swim meet, an official records the time it takes each swimmer to swim 100 meters for all swimmers who compete in only one stroke category. The following table shows the mean times and corresponding standard deviations for the collegiate women at the swim meet for each of the four stroke categories. For each of the 4 stroke categories, consider a random variable representing the time of a randomly selected swimmer in that category. What is the standard deviation of the sum of the 4 random variables?

• 0.83 seconds

• 1.67 seconds

• 2.80 seconds

• 3.32 seconds

• 3.76 seconds

Answer explanation

To find the standard deviation the variances are added. σ2 x+y+z+w = σ2X + σ2Y + σ2Z +σ2W = 0.70^2 + 0.92^2 + 0.94^2 + 0.76^2 = 2.7976, so σ=√ 2.7976 ~ 1.67.

A player pays $15 to play a game in which a chip is randomly selected from a bag of chips. The bag contains 10 red chips, 4 blue chips, and 6 yellow chips. The player wins $5 if a red chip is selected, $10 if a blue chip is selected, and $20 if a yellow chip is selected. Let the random variable X represent the amount won from the selection of the chip, and let the random variable WW represent the total amount won, where W=X−15. What is the mean of W ?

• $10.50

• $4.50

• -$4.50

• -$6.50

• -$10.50

Answer explanation

The expected payout is E(X) = 10/20($5) + 4/20($10) + 6/20($20) = $10.50. After subtracting the $15 to play the game, the result is –$4.50 expected winnings.

The distribution of random variable R has mean 10 and standard deviation 4. The distribution of random variable S has mean 7 and standard deviation 3. If R and S are independent, what are the mean and standard deviation of the distribution of R−S ?

• Mean 3 and standard deviation 1

• Mean 3 and standard deviation 5

• Mean 3 and standard deviation 7

• Mean 17 and standard deviation 1

• Mean 17 and standard deviation 5

Answer explanation

When subtracting two independent random variables, the means are subtracted and the variances are added. Thus, μR−S = μR−μS =1 0−7 =3, and σ^2R−S = σ^2R + σ^2S = 4^2 + 3^2 = 25, so σR−S=√25 = 5

At a certain company, loan agents are paid based on the number of loans they close in a day. Based on company records, the number of loans X that a randomly selected loan agent closes on a randomly selected day has the probability distribution below. At the company, the daily salary of a loan agent is $150 plus $50 per loan closed. Let Y represent the amount of money made by a randomly selected loan agent on a randomly selected day. Which of the following statements is NOT true?

• The mean of X is less than the mean of Y.

• The standard deviation of Y is approximately $71

• The mean daily salary is greater than $350 per day

• The standard deviation of X is less than the standard deviation of Y

• The shape of the probability distribution of Y is unimodal and roughly symmetric

Answer explanation

Since the mean of X is μx = 1(0.05) + 2(0.10) + 3(0.22) + 4(0.30) + 5(0.18) + 6(0.12) + 7(0.03) = 3.94, then the mean of Y is μY = a + bμX = 150 + 50(3.94) = $347. Since $347 < $350, the statement is false.

Data were collected on the ages, in years, of the men and women enrolled in a large sociology course. Let the random variables M and W represent the ages of the men and women, respectively. The distribution of M has mean 20.7 years and standard deviation 1.73 years. The distribution of W has mean 20.2 years and standard deviation 1.60 years. Of all of those enrolled in the course, 54 percent are men and 46 percent are women. What is the mean age of the combined distribution of both men and women in the course?

• 20.2 years

• 20.43 years

• 20.45 years

• 20.47 years

• 40.9 years

Answer explanation

Since 54% of the population are men and 46% are women, the mean of the combined distribution of men and women is 0.54M + 0.46W which is 0.54(20.7) + 0.46(20.2) = 20.47

The random variable X has mean 12 and standard deviation 3. The random variable WW is defined as W=7+2X. What are the mean and standard deviation of W?

• The mean is 24, and the standard deviation is 6.

• The mean is 24, and the standard deviation is 13.

• The mean is 31, and the standard deviation is 3.

• The mean is 31, and the standard deviation is 6.

• The mean is 31, and the standard deviation is 13.

Answer explanation

The mean and standard deviation of W are μW = a + bμX = 7 + 2(12) = 31 and σW = |b|σX = |2|(3) = 6, respectively. Mean is affected by the multiplication and addition of a constant, but standard deviation is affected only by the multiplication of a constant