Mean and Variance of an Exponential Distribution from MGF | UPSC ISS 2024 Paper-1 | Problem-40

Описание: Given the moment generating function (MGF) of a distribution:
\[
M_X(t) = \frac{\lambda}{\lambda - t} \quad \text{for } t \lambda,
\]
this corresponds to the exponential distribution with parameter \(\lambda\). For an exponential distribution, the expectation and variance are well-known:
\[
E[X] = \frac{1}{\lambda}, \quad \text{Var}(X) = \frac{1}{\lambda^2}.
\]

Therefore, the mean of the distribution is \(\frac{1}{\lambda}\), and the variance is \(\frac{1}{\lambda^2}\). The answer confirms that option B is correct, emphasizing that these fundamental moments can be directly derived from the MGF's form for the exponential distribution.

This explanation reinforces the link between MGFs and the moments of common probability distributions, which is essential for statistical inference and parameter estimation in real-world applications.

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