Question

Let \( X \) be a random variable having discrete uniform distribution on \( \{1, 3, 5, 7, \dots, 99\} \). Then \( E(X \mid X \text{ is not a multiple of 15}) \) equals

• A. \( \frac{2365}{47} \)
• B. \( \frac{2365}{50} \)
• C. 50
• D. 47

Hint:

For a uniform distribution, the expected value is the sum of the values divided by the number of values.

Answer 1

The Correct Option is A

Step 1: Understanding the distribution of \( X \).
The random variable \( X \) has a discrete uniform distribution on the set \( \{1, 3, 5, 7, \dots, 99\} \), which contains all odd numbers from 1 to 99. There are 50 elements in this set because the odd numbers between 1 and 99 are in the form \( 2n + 1 \), where \( n \) ranges from 0 to 49. Hence, the total number of odd numbers between 1 and 99 is 50.
Step 2: Calculating the sum of all possible values of \( X \).
The sum of the first 50 odd numbers is given by: \[ S = 1 + 3 + 5 + 7 + \dots + 99 \] The sum of the first \( n \) odd numbers is known to be \( n^2 \). Therefore: \[ S = 50^2 = 2500 \] Step 3: Identifying multiples of 15 in the set.
Next, we find the multiples of 15 in the set \( \{1, 3, 5, 7, \dots, 99\} \). The odd multiples of 15 are: \[ 15, 45, 75 \] Thus, there are 3 multiples of 15 in the set, and their sum is: \[ 15 + 45 + 75 = 135 \] Step 4: Subtracting the sum of multiples of 15.
The sum of all odd numbers excluding the multiples of 15 is: \[ \text{Sum excluding multiples of 15} = 2500 - 135 = 2365 \] Step 5: Finding the expected value of \( X \) given that \( X \) is not a multiple of 15.
The number of odd numbers that are not multiples of 15 is: \[ 50 - 3 = 47 \] Therefore, the expected value of \( X \) given that it is not a multiple of 15 is the average of the remaining values, which is: \[ E(X \mid X \text{ is not a multiple of 15}) = \frac{2365}{47} \]