GATE 2024 DA – Question 57

Probability and Statistics · Random variables, discrete and continuous distributions · 2 marks · Numerical answer

Let $X$ be a random variable exponentially distributed with parameter $\lambda > 0$. The probability density function of $X$ is given by: $f_X(x) = \lambda e^{-\lambda x}$ for $x \ge 0$ and 0 otherwise. If $5E(X) = Var(X)$, where $E(X)$ and $Var(X)$ indicate the expectation and variance of $X$, respectively, the value of $\lambda$ is ______ (rounded off to one decimal place).

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Correct answer: 0.19 to 0.21

Explanation

For the exponential distribution, $E(X) = \frac{1}{\lambda}$ and $Var(X) = \frac{1}{\lambda^2}$. The condition $\frac{5}{\lambda} = \frac{1}{\lambda^2}$ gives $\lambda = \frac{1}{5} = 0.2$.