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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 7: Q.28 (page 361)

Show that $Cov (X, E [Y ∣ X]) = Cov (X, Y)$ .

Short Answer

Expert verified

We prove that, $Cov (X, E [Y ∣ X]) = Cov (X, Y)$

Step by step solution

Given information

Given in the question that, we have to prove $Cov (X, E [Y ∣ X]) = Cov (X, Y)$

Explanation

Let's start from the left side. We have that

$= E (X Y) - E (X) E (Y)$

$= Cov (X, Y)$

Here we have used basic properties of conditional expectation.

Final answer

We prove that, $Cov (X, E [Y ∣ X]) = Cov (X, Y)$

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Most popular questions from this chapter

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

In Example 5c, compute the variance of the length of time until the miner reaches safety.

Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let W denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win. Find

(a) P{W > 0}

(b) P{W < 0}

The joint density of $X$ and $Y$ is given by

$f (x, y) = \frac{1}{\sqrt{2 π}} e^{- y} e^{- (x - y)^{2} / 2} 0 < y < \infty$ ,

$- \infty < x < \infty$

(a) Compute the joint moment generating function of $X$ and $Y$ .

(b) Compute the individual moment generating functions.

We say that $X$ is stochastically larger than $Y$ , written $X \geq_{st} Y$ , if, for all $t$ ,

$P {X > t} \geq P {Y > t}$

Show that if $X \geq_{st} Y$ then $E [X] \geq E [Y]$ when

(a) $X$ and $Y$ are nonnegative random variables;

(b) $X$ and $Y$ are arbitrary random variables. Hint:

Write $X$ as

$X = X^{+} - X^{-}$

where

$X^{+} = \{\begin{array}{ll} X if X \geq 0 \\ 0 if X < 0 \end{array}, X^{-} = \{\begin{aligned} 0 if X \geq 0 \\ - X if X < 0 \end{aligned}$

Similarly, represent $Y$ as $Y^{+} - Y^{-}$ . Then make use of part (a).

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Show that $Cov (X, E [Y ∣ X]) = Cov (X, Y)$ .

Short Answer

Step by step solution

Given information

Explanation

Final answer

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Most popular questions from this chapter

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Show thatCov(X,E[Y∣X])=Cov(X,Y).

Short Answer

Step by step solution

Given information

Explanation

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Statistics

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Show that $Cov (X, E [Y ∣ X]) = Cov (X, Y)$ .