Chapter 7: Q.7.10 (page 359)

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed positive random variables. For $k \leq n$ find $E [\frac{\sum_{i = 1}^{k} X_{i}}{\sum_{i = 1}^{n} X_{i}}] .$

Short Answer

Expert verified

The value of $E [\frac{\sum_{i = 1}^{k} X_{i}}{\sum_{i = 1}^{n} X_{i}}]$ is $\{\begin{cases} 0, n = k \\ \frac{k}{n}, n > k \end{cases} .$

Step by step solution

Given Information

Independent and identically distributed positive random variables are $X_{1}, X_{2}, \dots, X_{n}$ .

Find $E [\frac{\sum_{i = 1}^{k} X_{i}}{\sum_{i = 1}^{n} X_{i}}] = ?$

Explanation

It is formally perceived that the sum of the independent and identically distributed random variables approaches a normal distribution with the mean $μ$ and the standard deviation $σ$ . As given in the question:

$E [\frac{\sum_{i = 1}^{k} X_{i}}{\sum_{i = 1}^{n} X_{i}}] = \frac{E \sum_{i = 1}^{k} X_{i}}{E \sum_{i = 1}^{n} X_{i}}$

$= \frac{E [X_{1} + X_{2} + \dots \dots + X_{k}]}{E [X_{1} + X_{2} + \dots \dots + X_{n}]}$

Explanation

The expectation of sums of random variables is always equals to the sum of the expectation of each random variable.

If $E [X_{i}] = μ$ (finite)then:

$\frac{E [X_{1} + X_{2} + \dots \dots + X_{k}]}{E [X_{1} + X_{2} + \dots \dots + X_{n}]} = \frac{E [X_{1}] + E [X_{2}] + \dots \dots + E [X_{k}]}{E [X_{1}] + E [X_{2}] + \dots \dots + E [X_{n}]}$

$= \frac{μ + μ + \dots \dots + μ (k times)}{μ + μ + \dots \dots + μ (n times)}$ where $n > k$

$= \frac{k μ}{n μ}$

$= \frac{k}{n}$

Therefore:

$E [\frac{\sum_{i = 1}^{k} X_{i}}{\sum_{i = 1}^{n} X_{i}}] = \{\begin{cases} 0, n = k \\ \frac{k}{n}, n > k \end{cases}$

Final Answer

Hence, the value of $E [\frac{\sum_{i = 1}^{k} X_{i}}{\sum_{i = 1}^{n} X_{i}}]$ is $\{\begin{cases} 0, n = k \\ \frac{k}{n}, n > k \end{cases} .$

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Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed positive random variables. For $k \leq n$ find $E [\frac{\sum_{i = 1}^{k} X_{i}}{\sum_{i = 1}^{n} X_{i}}] .$

Short Answer

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Let X1,X2,…,Xn be independent and identically distributed positive random variables. For k≤n findE∑i=1kXi∑i=1nXi.

Short Answer

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Explanation

Final Answer

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Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed positive random variables. For $k \leq n$ find $E [\frac{\sum_{i = 1}^{k} X_{i}}{\sum_{i = 1}^{n} X_{i}}] .$