# Let X be a discrete random variable with mean mu and variance sigma^2 . Answer these: ...

## Question:

Let {eq}X {/eq} be a discrete random variable with mean {eq}\mu {/eq} and variance {eq}\sigma^2 {/eq}. Answer these:

(a) For {eq}n {/eq} independent draws {eq}X_1, ..., X_n {/eq} let {eq}\bar{X} = \sum_{i = 1}^n Xi {/eq} . What is the mean and variance of {eq}\bar{X} {/eq}

(b) Let {eq}Y = aX + b {/eq} with {eq}a, b {/eq} constants. What is the mean and variance of for the average of {eq}n {/eq} independent draws, {eq}Yi = aXi + b {/eq}

(c) What is {eq}E(Y |X = x) {/eq}

(d) Now, let {eq}X {/eq} be continuous with the same mean and variance. Answer (a)-(c) again.

(f) Redefine the random variable as {eq}X\sim Ber(p) {/eq}

## Expectation

The Expectation is simply an average value. The mean value or the expected value, both are the same but are used in different contexts. When talking about the probability distribution, the mean value is used.

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Given Information

{eq}E\left( X \right) = \mu \;,\;Var\left( X \right) = {\sigma ^2} {/eq}

(a)

{eq}\begin{align*} \tilde X &= \sum\limits_i^n...

Evaluating the Expected Value of Sample Information

from

Chapter 5 / Lesson 10
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Many companies consider doing surveys and market research but are not sure whether the expense is worth the data that will be collected. This lesson helps explain how to calculate the expected value of sample information.