Let $ \{X_n\}_{n \geq 1} $ be a sequence of independent and identically distributed random variables each having probability density function $$ f(x) = \begin{cases} e^{-x}, & x0 \ 0, & \text{otherwise} \end{cases} $$ Let $ X_{(n)} = \max\{ X_1, X_2, \dots, X_n \} $ for $ n \geq 1 $. If $ Z $ is the random variable to which $$ \{ X_{(n)} - \log n \}_{n \geq 1} $$ converges in distribution, as $ n \to \infty $, then the median of $ Z $ equals $$ \underline{\hspace{2cm}} $$ (round off to 2 decimal places).

Question

Let $ \{X_n\}_{n \geq 1} $ be a sequence of independent and identically distributed random variables each having probability density function $$ f(x) = \begin{cases} e^{-x}, & x>0 \ 0, & \text{otherwise} \end{cases} $$ Let $ X_{(n)} = \max\{ X_1, X_2, \dots, X_n \} $ for $ n \geq 1 $. If $ Z $ is the random variable to which $$ \{ X_{(n)} - \log n \}_{n \geq 1} $$ converges in distribution, as $ n \to \infty $, then the median of $ Z $ equals $$ \underline{\hspace{2cm}} $$ (round off to 2 decimal places).

cdquestions Admin · Accepted Answer

To find the median of $ Z $, we examine the behavior of $ X_{(n)} $ and the limiting distribution of $ Z $ as $ n \to \infty $. Using standard results for the distribution of the maximum of independent identically distributed random variables, we find that the median of $ Z $ is approximately: $$ \text{Median}(Z) \approx 0.32. $$ Thus, the value is $ 0.32 $.

Show Hint

Correct Answer: 0.32

Solution and Explanation

Top Questions on Random Variables

Questions Asked in GATE ST exam