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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.7.33 (page 354)

If $E [X] = 1$ and $V a r (X) = 5$ find
(a) $E [(2 + X^{2})]$
(b) $V a r (4 + 3 X)$

Short Answer

Expert verified

a) The value of $E [(2 + X^{2})]$ is $14$ .

b) The value of $Var (4 + 3 X)$ is $45 .$

Step by step solution

01

Given information (Part a)

$E (X) = 1$

$V a r (X) = 5$

Then $E [(2 + X^{2})] = ?$

02

Explanation (Part a)

We know that

$E (X^{2}) = Var (X) + (E (X))^{2}$

$= 5 + 1$

$= 6$

Calculate the expected value of $E [(2 + X^{2})]$

$E [(2 + X)^{2}] = E [4 + X^{2} + 4 X]$

$= 4 + 6 + 4$

$= 14$

03

Final Answer (Part a)

The value of $E [(2 + X^{2})]$ is $14 .$

04

Given information (Part b)

$E (X) = 1$

$Var (X) = 5$

Then $Var (4 + 3 x) = ?$

05

Explanation (Part b)

We know that,

$E (X^{2}) = Var (X) + (E (X))^{2}$

$= 5 + 1$

role="math" localid="1647519062218" $= 6$

Calculate the variance.

$V a r (4 + 3 X) = 0 + 9 V a r (X)$

$= (9 \times 5)$

$= 45$

06

Final Answer (Part b)

Hence, the value of $V a r (4 + 3 x)$ is $45 .$

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Most popular questions from this chapter

A prisoner is trapped in a cell containing $3$ doors. The first door leads to a tunnel that returns him to his cell after $2$ days’ travel. The second leads to a tunnel that returns him to his cell after $4$ days’ travel. The third door leads to freedom after $1$ day of travel. If it is assumed that the prisoner will always select doors $1, 2,$ and $3$ with respective probabilities $. 5, . 3,$ and . $2$ , what is the expected number of days until the prisoner reaches freedom?

Show that $E [{(X - a)}^{2}]$ is minimized at $a = E [X]$ .

Cards from an ordinary deck of $52$ playing cards are turned face upon at a time. If the 1st card is an ace, or the $2$ nd a deuce, or the $3$ rd a three, or ...,or the $13$ th a king,or the $14$ an ace, and so on, we say that a match occurs. Note that we do not require that the ( $13$ n + $1$ ) card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.

7.2. Suppose that $X$ is a continuous random variable with

density function $f$ . Show that $E [I X - a ∣]$ is minimized

when $a$ is equal to the median of $F$ .

Hint: Write

$E [I X - a l] = | x - a | f (x) d x$

Now break up the integral into the regions where $x < a$

and where $x > a$ , and differentiate.

The random variables X and Y have a joint density function is given by

$f (x, y) = {\begin{cases} 2 e^{- 2 x} / x & 0 \leq x < \infty, 0 \leq y \leq x \\ 0 & otherwise \end{cases}$

Compute $Cov (X, Y)$

See all solutions

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