Stochashic variable result(Please verify) Urgentby Beowulf2007 Tags: resultplease, stochashic, urgent, variable, verify 

#1
Dec1207, 01:14 PM

P: 18

1. The problem statement, all variables and given/known data
Dear All, I have this problem here. Part(1) 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 Proof 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] Proof 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. BR Beowulf.... 


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