ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 20 - 25 of 39

Let X be exponentially distributed with parameter θ.

The likelihood function is

The log-likelihood function is

Differentiable with respect to θ, equating to zero

When population mean and population standard deviation in not know. If is the samplemean and replace σ by its estimate s and t _α/2 be the critical value of the student t-test such that have of the area on the left hand side and other half on the right side that is

For first inequality we get

From secon

By using factorization theorem, the condition is that

where h (x) is free from θ and g (. ) depends on X only through T.

(i)

The joint pdf of random sample is

Let T= . By factorization theorem

h (x) =1,

So, h (x) is free from θ

T ² statistic can be computed from X. Here we have to compute it form Y and both will be same. Here

Let , then and

So, T ² -statistic comes into existence for testing hypothesis of the form

Since is known vector,

If we have a setup

Let X= (X ₁, …, X _k) and also let E _k be the set of all points in k dimensional Euclidean space R _k, for which we reject H ₀ using the sequential probability ratio test. Also, let F _k be the set of all points in R _k for which we accept H ₀. Notice that (E _k, k = 1,2, …) are mutually disjoint and (F _k, k = 1,2, …) are also mutually disjoint. Assu

(i) the distribution of 3X ₁ -2X ₂ +X ₃ is

The mean is

The variance-covariance matrix is