Stochashic variable result(Please verify) Urgent

by Beowulf2007
Tags: resultplease, stochashic, urgent, variable, verify
Beowulf2007 is offline
Dec12-07, 01:14 PM
P: 18
1. The problem statement, all variables and given/known data

Dear All,

I have this problem here.


Let Y be a stochastic variable with the distribution function [tex]F_{Y}[/tex] given by:

[tex]P(Y \leq y) = F_{Y}(y) = \left\{ \begin{array}{ccc} \ 0& \ \ \mathrm{if} \ y < 0 \\ sin(y)& \ \ \mathrm{if} \ y \in [0,\pi/2] \\ 1& \ \ \mathrm{if} \ y > 0. \end{array}[/tex]

Explain why Y is absolute continious and give the density [tex]f_{Y}.[/tex]

3. The attempt at a solution


Using the following theorem:

Let X be a stochastic variable with the distribution function F, Assuming that F is continuous and that F' exists in all but finite many points [tex]x_{1} < x_{2} < x_\ldots < x_{n}[/tex]. Then X is absolutely continuous with the desensity

[tex]f(x) = \left\{ \begin{array}{ccc} F'(x) \ \ &\mathrm{if} \ \ x \notin \{x_{1}, x_{2}, \ldots, x_{n} \} \\ 0 \ \ &\mathrm{if} \ \ x \in \{x_{1}, x_{2}, \ldots, x_{n} \}. \end{array}[/tex]

By the theorem above its clearly visable that [tex]F_Y[/tex] is continous everywhere by the definition of continouty, then F' exists and thusly

[tex]f_{Y} = \frac{d}{dy}(sin(y)) = cos(y).[/tex]

Therefore Y is absolute continious.

Part two

Let X be a absolute continous stochastic variable with the probability density [tex]f_{X}[/tex] given by

[tex]f_{X}(x) = \left\{ \begin{array}{ccc} \frac{1}{9}|x|& \ \ \mathrm{if} \ \ x \in ]-3,3[ \\ 0& \ \ \mathrm{otherwise.} \end{array}[/tex]

Show that [tex]P(|X| \leq 1) = \frac{1}{9}.[/tex]


Since we know that the density function is given according to the definition

[tex]\int_{-\infty}^{\infty} f(x) dx = \int_{-3}^{3} \frac{1}{9}|x| dx = 1[/tex] Then to obtain where [tex]P(|X| \leq 1)[/tex] we analyse the interval

[tex]x \in \{-1,1\}[/tex] from which we obtain

[tex]P(|X| \leq 1) = \int_{-1}^{1} \frac{1}{9}|x| dx = \frac{1}{9} \cdot \int_{-1}^{1} |x| dx = [\frac{x \cdot |x|}{18}]_{x=-1}^{1} = \frac{1}{9}.[/tex]

How does part one and two look? Do I need to add more text if yes what?

The problem is correctly formulated from my textbook.

Thanks in advance.

Phys.Org News Partner Science news on
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes

Register to reply

Related Discussions
Urgent Magnetic Field Problem!!! Urgent!!!!!!!!!!please Help! Introductory Physics Homework 14
forces problem(urgent urgent help needed) Introductory Physics Homework 3
Quantum Mechanics Homework Help(Urgent Urgent Urgent)! Advanced Physics Homework 3
Euler: Please verify my result ! Calculus & Beyond Homework 0
Urgent: Linear Algebra Question(Please verify) Calculus & Beyond Homework 1