Question

1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 < θ< I. We if and only if Σ x, > 4 G) Find the Type I error probability of this test. (ii) Obtain the power function and construct the power curve of this test. (c) Xi, X, , X, is a random sample from the Gamma (α, λ) distribution. Let x = Σ X, . What is the distribution of X ? /n

Answer 1

b)Probability of type I error =P(Rejceting null hypothesis/ null is true)

$=P(\sum_i^5X_i>4/\theta=1/2)$

$\sum_i^5X_i$ has binomial distribution with n=5 and p=1/2 under the null hypothesis

Hence the above probability is $\binom{5}{5}(1/2)^5 (1/2)^0$ =0.03125

Power function of the test is

$=P(\sum_i^5X_i>4/\theta>1/2)=\binom{5}{5}(\theta)^5 (1-\theta)^0=\theta^5; \theta>1/2$

c) The m.g.f of gamma distribution with $(\alpha,\lambda)$ is $(1-\lambda t)^{-\alpha}$

$M_{\bar{X}}(t)=M_{\sum X_i}(t/n)=\prod_{i=1}^{n}M_{X_i}(t/n)=\prod_{i=1}^{n}(1-\frac{\lambda t}{n})^{-\alpha}=(1-\frac{\lambda t}{n})^{-n \alpha}$ which is a mgf of gamma distribution with parameters $(n \alpha,\lambda/n)$