) Let X1 be an observation from an exponential distribution with

Question

) Let X₁ be an observation from an exponential distribution with the p.d.f.

$$f(x) = \frac{1}{\theta} e^{-x/\theta}; x > 0$
Test the null hypothesis that the mean of the distribution is $\theta = 2$ against the alternative hypothesis that is $\theta = 5$ . The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. Find the probabilities of type-I and type-II errors.

09 Jan 2026

Answer :

Word Count : 262

We are asked to find the probabilities of type-I and type-II errors for a hypothesis test involving an exponential distribution. Let's solve it step by step. We are given: * (X_1 \sim \text{Exponential}(\theta)), with p.d.f. (f(x) = \frac{1}{\theta} e^{-x/\theta}, x>0) * Null hypothesis _________ ____ ___ _______ ____ ___ _______ ___ _______ _____ __________ __________.
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