Please do all parts and please show step by step Let the sample space $ S $ be the triangle with corners $(0,0)$, $(1,0)$, $(0,1)$ with a uniform probability measure. Define random variables $ X $ and $ Y $ on $ S $ by: $ X((x,y)) = x $ and $ Y((x,y)) = y $.a. Find $ f_{XY}(x, y) $b. Find $ f_{X|Y}(x|y) $c. Find $ E[X|Y = y] $ (your answer will be a function of $ y $)d. Find $ Var[X|Y = y] $ (your answer will be a function of $ y $)e. Find $ f_Y(y) $f. Find $ E[Var[X|Y]] $g. Find $ Var[E[X|Y]] $

Answered: Let the sample space S be the triangle…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Please do all parts and please show step by step

Let the sample space $ S $ be the triangle with corners $(0,0)$, $(1,0)$, $(0,1)$ with a uniform probability measure. Define random variables $ X $ and $ Y $ on $ S $ by: $ X((x,y)) = x $ and $ Y((x,y)) = y $.

a. Find $ f_{XY}(x, y) $

b. Find $ f_{X|Y}(x|y) $

c. Find $ E[X|Y = y] $ (your answer will be a function of $ y $)

d. Find $ Var[X|Y = y] $ (your answer will be a function of $ y $)

e. Find $ f_Y(y) $

f. Find $ E[Var[X|Y]] $

g. Find $ Var[E[X|Y]] $

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Step 1: Given Information:

Note: “Since you have posted a question with multiple sub parts, we will provide the solution only to the first three sub parts as per our Q&A guidelines. Please repost the remaining sub parts separately.”

The area of the triangle formula is,

$A r e a equals 1 half b h space b equals b a s e h equals h e i g h t$

The formula to calculate $f subscript X vertical line Y end subscript left parenthesis x vertical line y right parenthesis$ is,

$f subscript X vertical line Y end subscript left parenthesis x vertical line y right parenthesis equals fraction numerator f subscript X Y end subscript left parenthesis x y right parenthesis over denominator f subscript Y left parenthesis y right parenthesis end fraction$

The conditional expectation of X|Y=y is ,

$E left parenthesis X vertical line Y equals y right parenthesis equals integral subscript X space x asterisk times f subscript X vertical line Y end subscript left parenthesis x vertical line y right parenthesis space d x$

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