Page 210 - Probability and Statistical Inference

4. Functions of Random Variables and Sampling Distribution 187

Obviously, G(u) = 0 for u ≤ 0 or u ≥ 1. With 0 < u < 1, we write

Thus, the pdf of U is given by

We leave out some of the intermediate steps as the Exercise 4.2.12. !
The convolution approach leads to the distribution of U = X + X
1
2
where X , X are iid Uniform(0, 1). The random variable U is often
2
1
said to have the triangular distribution. See the Exercise 4.2.9.
4.2.5 The Sampling Distribution
Suppose that we consider a large population of all adult men in a city and we
wish to gather information regarding the distribution of heights of these indi-
th
viduals. Let X stand for the height of the j individual selected randomly from
j
the whole population, j = 1, ..., n. Here, n stands for the sample size. Since the
population is large, we may assume from a practical point of view, that these
n individuals are selected independently of each other. The simple random
sampling with replacement (see the Example 1.7.6) would certainly gener-
ate independence. What is the sampling distribution of, say, the sample mean,
? How can we proceed to understand what this theoretical
distribution of may possibly mean to us in real life? A simple minded ap-
proach may consist of randomly selecting n individuals from the relevant
population and record each individual’s height. The data would look like n
numbers X , ..., X which will give rise to an observed value , namely
1,1
1,n
From the same population, if we contemplate selecting an-
other batch of n individuals, drawn independently from the first set, and record
their heights, we will end up with another data of n numbers x , ..., x leading
2,1
2,n
to another observed value of the sample mean , namely .
Though we do not physically have to go through this process of independent resampling
with the same sample of size n, we may think of such a hypothetical process of the
identical replication, infinitely many times. Through these replications, we will finally