Exercise 1 (Exponential distribution). Let X ~ Exp(1), that is, X has density function p(x) = e¯*1{x20}- 1. Calculate the mean and the the second moment of X. You can use the integration by parts formula. 2. Calculate E[X-1/2]. You can use a Gaussian integral formula.
Exercise 1 (Exponential distribution). Let X ~ Exp(1), that is, X has density function p(x) = e¯*1{x20}- 1. Calculate the mean and the the second moment of X. You can use the integration by parts formula. 2. Calculate E[X-1/2]. You can use a Gaussian integral formula.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Exercise 1 (Exponential distribution). Let X ~ Exp(1), that is, X has density function p(x) = e¯×1,
1. Calculate the mean and the the second moment of X. You can use the integration by parts formula.
2. Calculate E[X-1/2]. You can use a Gaussian integral formula.
{x20}•
→ (R, B(R)) which is bounded in L².
Exercise 2 (Cantelli's inequality). Consider a random variable X : (2, F, P)
Denote by u = B[X] its average and o2 = E[(X – µ)²] its variance.
1. Show that for all A > 0 and u > 0,
o2 +u?
E[(X – µ + u)²]
(A + u)2
P[X - μ > λ] <
(а+ и)?
2. Deduce Cantelli's inequality:
{
if a > 0;
> 1- if A < 0.
PIX - μ Σ λ]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F191ce871-8f03-4cb7-9658-a3fafde3334d%2F03a8e6f2-351f-4217-8525-50467c918e32%2Frfvgp5e_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 1 (Exponential distribution). Let X ~ Exp(1), that is, X has density function p(x) = e¯×1,
1. Calculate the mean and the the second moment of X. You can use the integration by parts formula.
2. Calculate E[X-1/2]. You can use a Gaussian integral formula.
{x20}•
→ (R, B(R)) which is bounded in L².
Exercise 2 (Cantelli's inequality). Consider a random variable X : (2, F, P)
Denote by u = B[X] its average and o2 = E[(X – µ)²] its variance.
1. Show that for all A > 0 and u > 0,
o2 +u?
E[(X – µ + u)²]
(A + u)2
P[X - μ > λ] <
(а+ и)?
2. Deduce Cantelli's inequality:
{
if a > 0;
> 1- if A < 0.
PIX - μ Σ λ]
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