Tazmin Horton

2021-11-04

Assume that X and Y are jointly continuous random variables with joint probability density function given by

Find E[X] and E[Y].

gotovub

Skilled2021-11-05Added 98 answers

Step 1

$E\left[X\right]={\mu}_{X}$

$={\int}_{x}xg\left(x\right)dx$

$={\int}_{0}^{2}x\left(\frac{1}{18}(x+8)\right)dx$

$=\frac{1}{18}{\int}_{0}^{2}({x}^{2}+8x)dx$

$=\frac{1}{18}{[\frac{{x}^{3}}{3}+8\frac{{x}^{2}}{2}]}_{0}^{2}$

$=\frac{1}{18}[\frac{8}{3}+8\left(2\right)]$

$=\frac{8}{18}\left[\frac{7}{3}\right]$

$=\frac{56}{54}$

$=\frac{28}{27}$

Step 2

E[Y]:

${\mu}_{Y}={\int}_{y}yh\left(y\right)dy$

$=\frac{1}{6}y\left(\frac{1}{6}(y+1)\right)dy$

$=\frac{1}{6}\left(\frac{52+24}{6}\right)$

$=\frac{1}{6}\left[\frac{16+24}{6}\right]$

$=\frac{76}{36}$

$=\frac{19}{9}$

Step 2

E[Y]:

22+64

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