Median Exceeding a Threshold

Solution & Explanation

To solve this problem, we need to determine the probability that the median of three independent and identically distributed random variables, drawn from a uniform distribution on the interval $[0, 4]$, exceeds 3.

Step-by-Step Approach:

1. Understanding the Uniform Distribution:

   • Each random variable, say $X_i$, follows a uniform distribution over $[0, 4]$. This implies that the probability density function (pdf) is constant and equal to $\frac{1}{4}$ over this range.

   • The probability of a single variable being greater than 3 is calculated as:

     $P(X_i > 3) = \frac{4 - 3}{4} = \frac{1}{4}.$

2. Defining the Events:

   • Event A: All three variables are greater than 3.
   • Event B: Two variables are greater than 3, and one is less than or equal to 3.
   • The median is defined as the middle value when the variables are sorted in ascending order. For the median to be greater than 3, at least two of the variables must be greater than 3.

3. Calculating Probabilities:

   • Probability of Event A:

     • All three variables exceed 3:

     $P(A) = \left(\frac{1}{4}\right)^3 = \frac{1}{64}.$

   • Probability of Event B:

     • Two variables exceed 3, and one does not. The probability for a single arrangement (e.g., $X_1 > 3, X_2 > 3, X_3 \leq 3$) is:

     $P(X_1 > 3, X_2 > 3, X_3 \leq 3) = \left(\frac{1}{4}\right)^2 \cdot \left(\frac{3}{4}\right) = \frac{3}{64}.$

     • There are 3 such arrangements (since any of the three variables can be the one that is less than or equal to 3):

     $P(B) = 3 \cdot \frac{3}{64} = \frac{9}{64}.$

4. Total Probability:

   • The probability that the median exceeds 3 is the sum of the probabilities of events A and B:

     $P(\text{Median} > 3) = P(A) + P(B) = \frac{1}{64} + \frac{9}{64} = \frac{10}{64} = \frac{5}{32}.$

Conclusion:

The probability that the median of three independent and identically distributed random variables drawn from a uniform distribution over $[0, 4]$ exceeds 3 is $\frac{5}{32}$. This solution considers all possible configurations of the variables that satisfy the condition of the median being greater than 3, ensuring a comprehensive and accurate calculation.