Q.5.11 Let Z be a standard normal rando... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 5: Q.5.11 (page 215)

Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g'.
(a) Show that E[g'(Z)]=E[Zg(Z)];
(b) Show that E[Z^{n+ $1$}]=nE[Z^{n- $1$}].
(c) Find E[Z^$4$].

Short Answer

Expert verified

Showing that ,

a) $E [Z g (z)] = {鈭�}_{鈭� 鈭�}^{鈭�} Z g (z) F (z) d z$

$= {鈭�}_{鈭� 鈭�}^{鈭�} g^{鈥�} (2) F (z) d z$

b) $E (z^{n + 1}) = {鈭�}_{鈭� 鈭�}^{鈭�} z^{n + 1} F (2) d z$

$= {鈭�}_{鈭�}^{鈭�} n z^{n 鈭� 1} F (z) d z$

$= n {鈭�}_{鈭� 鈭�}^{鈭�} z^{n 鈭� 1} F (n) d z +$

c) $E (z^{4}) = 3$

Step by step solution

Given Information (part a)

Show that E[g'(Z)]=E[Zg(Z)]

Step 2: Explanation (part a)

Let us Consider ' Z " to be a standard normal random variable z a let 'g' be the differentiable function with derivative g.

$We need to show E (g (z)) = E [z_{g} (z)]$

$E [Z g (z)] = {鈭�}_{鈭� 鈭�}^{鈭�} Z g (z) F (z) d z$

$= {鈭�}_{鈭� 鈭�}^{鈭�} g^{鈥�} (2) F (z) d z$

$E [z g (z)] = E [g (z)]$

Final Answer (part a)

Showing that,

$E [Z g (z)] = {鈭�}_{鈭� 鈭�}^{鈭�} Z g (z) F (z) d z$

$= {鈭�}_{鈭� 鈭�}^{鈭�} g^{鈥�} (2) F (z) d z$

Given Information (part b)

Show that E[Z^{n+ $1$}]=n E[Z^{n- $1$}].

Explanation (part b)

$Now, E (z^{n + 1}) = {鈭�}_{鈭� 鈭�}^{鈭�} z^{n + 1} F (2) d z$

$= {鈭�}_{鈭�}^{鈭�} n z^{n 鈭� 1} F (z) d z$

$= n {鈭�}_{鈭� 鈭�}^{鈭�} z^{n 鈭� 1} F (n) d z +$

$E [z^{n + 1}] = n 鈭� [z^{n 鈭� 1}]$

Final Answer (part b)

shows that,

$E (z^{n + 1}) = {鈭�}_{鈭� 鈭�}^{鈭�} z^{n + 1} F (2) d z$

$= {鈭�}_{鈭�}^{鈭�} n z^{n 鈭� 1} F (z) d z$

$= n {鈭�}_{鈭� 鈭�}^{鈭�} z^{n 鈭� 1} F (n) d z +$

Given Information (part c)

Find E[Z^$4$].

Explanation (part c)

$Now find E (z^{4})$

$E (z^{4}) = E (z^{3 鈭� 1})$

$= 3 E (z^{2})$

$= 3 {vari 鈦� (z) + (E (z) r)}$

$= 3 [1 + 0]$

$= 3 (1)$

$E (z^{4}) = 3$

Final Answer (part c)

$E (z^{4}) = 3$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g'.
(a) Show that E[g'(Z)]=E[Zg(Z)];
(b) Show that E[Z^{n+ $1$}]=nE[Z^{n- $1$}].
(c) Find E[Z^$4$].

Short Answer

Step by step solution

Given Information (part a)

Step 2: Explanation (part a)

Final Answer (part a)

Given Information (part b)

Explanation (part b)

Final Answer (part b)

Given Information (part c)

Explanation (part c)

Final Answer (part c)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g'.(a) Show that E[g'(Z)]=E[Zg(Z)];(b) Show that E[Zn+1]=nE[Zn-1].(c) Find E[Z4].

Short Answer

Step by step solution

Given Information (part a)

Step 2: Explanation (part a)

Final Answer (part a)

Given Information (part b)

Explanation (part b)

Final Answer (part b)

Given Information (part c)

Explanation (part c)

Final Answer (part c)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g'.
(a) Show that E[g'(Z)]=E[Zg(Z)];
(b) Show that E[Z^{n+ $1$}]=nE[Z^{n- $1$}].
(c) Find E[Z^$4$].