) 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 (H_0: \theta = 2) * Alternative hypothesis (H_1: \theta = 5) * Decision rule: accept (H_0) if (X_1 < 3), otherwise reject ____ ___ ___ ______ _________ __________ _____ ____ ________ __________ ________ _________.
_________ ___ ____ ________ _______ ________ _______.
______ _______ ___ __________ __________ _______ _________ ______ __________ ____.
____ _________ __________ ________ _________ ____ ___ ___ _________.
___ _________ ____ ____ _________ _________ ___ __________ _____ ______.
___ ____ ____ ____ ________ ___ _________ __________ _____ _______ __________.
___ ______ ___ ________ _______.
______ ___ _____ ____ _______.
___ __________ ____ ____ _____ _____ _____ ___ ___ ______ ____ _____.
____ ________ __________ ___ ____ _____ _______ _____ _______ _____.
_________ _________ ____ ________ _________ __________ ____ ________.
_____ ________ ___ ______ ________ ________ ________.
_________ __________ ______ ________ __________ ______.
_________ ___ ______ ___ _______ __________ ___ ______ _________ _____ ______ ____.
_____ _______ ______ ______ _________ ______ ________ _____ _______ _______ _________.
__________ __________ ________ ______ ______.
_____ ___ ______ _____ ______ _______ ______ ____ ____ ___.
___ _____ _______ ______ ___ _____ ___.
________ ________ __________ ______ __________ __________ ______ _________ ____ _______.
____ _________ __________ _________ __________ _________ ________ ___ ___.
___ ________ _____ ______ __________ ___.
_____ ______ ________ _________ ____ __________ _______ __________ ____ _________.
______ _________ ___ _____ _____.
Get Full Answer on WhatsApp