Sampling Distribution And Approximations Of Distributions

INTRODUCTION

A random variable is a function defined on a sample space. I follows that a function of a random variable is also a function defined over the same sample space. Hence a function of a random variable is itself a random variable and is associated with a probability distribution. In this Unit, our concern will be to determine the probability distribution of functions of one or more random variation.

We shall also assume certain approximations to probability distributions which has immense practical value.

Here, we describe certain methods to determining the probability distribution of a function of random variables and use them to find distributions of sums of some of the standard discrete and continuous random variables. This section introduces you also to probability distributions of certain functions of independently and identically distributed (id) normal random variables that are frequently used in statistical theory of Estimation and Tests of Hypotheses which you are going to study.

The concepts of the random samples and the sampling distribution of a statistic are introduced here. It establishes the relationship between the statistics to be used in Block 3 for drawing statistical inference and the distributions of functions of iid normal variables considered.

It is not always easy to derive and/or use the exact probability distribution of a function of random variables. In that a case, good approximations either to the concerned cdf or even to the probability of certain relevant events, serve the purpose of knowing the distribution. Here, we state, explain and also prove in one case, certain limit theorems on probability which form the basis of such approximations.

Here, we study, approximations of standard distributions like binomial, Poisson and hypergeometric by normal distribution. These approximations are of great practical value.