Question

From the set \[ \{2,3,5,7\} \] two numbers are selected one after the other with replacement. If \(X\) is the random variable representing the absolute difference of the two numbers selected, then the mean of \(X\) is

• A. \(\dfrac{15}{4}\)
• B. \(\dfrac{17}{8}\)
• C. \(\dfrac{18}{5}\)
• D. \(6\) \bigskip

Hint:

For discrete random variables, \[ E(X)=\sum xP(X=x). \] When selections are made with replacement, first count all ordered outcomes, construct the probability distribution, and then calculate the expectation.

Answer 1

The Correct Option is B

Concept: Expected value \(E(X)=\sum x P(X=x)\). Enumerate all 16 ordered pairs with replacement, tabulate the absolute differences, and compute the weighted average.

Step 1: List all pairs \((a,b)\) from \(\{2,3,5,7\}\) and their \(|a-b|\): 0 appears for (2,2),(3,3),(5,5),(7,7); 1 for (2,3),(3,2); 2 for (3,5),(5,3),(5,7),(7,5); 3 for (2,5),(5,2); 4 for (3,7),(7,3); 5 for (2,7),(7,2).

Step 2: Probability distribution: P(0)=4/16=1/4, P(1)=2/16=1/8, P(2)=4/16=1/4, P(3)=2/16=1/8, P(4)=2/16=1/8, P(5)=2/16=1/8.

Step 3: Compute E(X). \(0\cdot\frac14 + 1\cdot\frac18 + 2\cdot\frac14 + 3\cdot\frac18 + 4\cdot\frac18 + 5\cdot\frac18 = 0 + \frac18 + \frac12 + \frac38 + \frac48 + \frac58 = \frac{1+4+3+4+5}{8} = \frac{17}{8}\).

Step 4: Write the final answer. \(\boxed{\frac{17}{8}}\)