Question

At a certain bakery, the price of each doughnut is $1.50. Let the random variable DD represent the number of doughnuts a typical customer purchases each day. The expected value and variance of the probability distribution of DD are 2.6 doughnuts and 3.6 (doughnuts)^2, respectively. Let the random variable PP represent the price of the doughnuts that a typical customer purchases each day. Which of the following is the standard deviation, in dollars, of the probability distribution of PP?

A) 1.5√(3.6)
B) √(1.5² * 3.6)
C) √(1.5 * 3.6)
D) 1.5√(2.6)
E) 1.5√(3.6)

Answer 1

The standard deviation of the probability distribution of PP in dollars is found by multiplying the standard deviation of the number of doughnuts by the price per doughnut. The correct formula for this calculation is √(1.5² * 3.6), making option B the correct answer.

Step-by-step explanation:

The question asks to determine the standard deviation, in dollars, of the probability distribution of the random variable PP, which represents the price of the doughnuts a typical customer purchases each day. From the information provided, we know that the expected value of the random variable DD (the number of doughnuts) is 2.6 doughnuts, and its variance is 3.6 (doughnuts)2, given the price of each doughnut is $1.50.

The standard deviation of the prices, which we'll represent as σP, can be found by multiplying the standard deviation of the quantity of doughnuts (σD) by the price of each doughnut. Since the variance of DD is 3.6, the standard deviation of DD, σD, is the square root of 3.6, which yields:

• σD = √3.6

Thus, the standard deviation of the price σP is:

• σP = $1.50 * σD = $1.50 * √3.6

When squared, $1.50 becomes 2.25, and multiplying this with the variance of 3.6 doughnuts gives a new variance in terms of dollars as 2.25 * 3.6. Therefore, the standard deviation in dollars is the square root of this product:

• σP = √(2.25 * 3.6)

The correct answer is B) √(1.52 * 3.6).