Question

Two distinct integers are chosen randomly from 5 consecutive integers. If the random variable x represents the absolute difference between them, then the mean and variance of x are, respectively,

• A. 1 and 3/ 2
• B. 2 and 5
• C. 1 and 3
• D. 2 and 1

Answer 1

The Correct Option is D

Expected Value and Variance of Absolute Differences in Five Consecutive Integers

Consider five consecutive integers, for example: 1, 2, 3, 4, 5. 

The random variable X represents the absolute difference between any two randomly chosen integers from this set.

Step 1: Calculating Possible Differences

Possible absolute differences between any two chosen numbers are:

• |1 - 2| = 1, |1 - 3| = 2, |1 - 4| = 3, |1 - 5| = 4
• |2 - 3| = 1, |2 - 4| = 2, |2 - 5| = 3
• |3 - 4| = 1, |3 - 5| = 2
• |4 - 5| = 1

All possible differences and their probabilities are determined by counting occurrences.

Step 2: Calculating the Mean (E(X))

After computing the probability distribution of X, the expected value (mean) is found to be:

E(X) = 2

Step 3: Calculating the Variance (Var(X))

Using the variance formula:

Var(X) = E(X²) - (E(X))²

After calculations, the variance of X is determined to be:

Var(X) = 1