Question

Question 2. (20 pts.) The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions. 3. Evaluate Ply < x <0) 4. Find the correlation coefficient between X and Y having the joint density functions: f(x,y) »> = {ke***** for y2 + x? <4 elsewhere
The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions. 3. Evaluate ?(? < ? < 0) 4. Find the correlation coefficient between X and Y having the joint density functions:(.) ?(?,?) = {???2+?2 ??? ?2 + ?2 < 4 0 ?????h???

Answer 1

1) Area under PDF is always unity hence by using this property we can find k, therefore

$\int ke^{y^{2}+x^{2}}=1$

let x't you Skelry and POF is always positive and it is given sc²+ y² L 4 3 u limits are oto 3 Ske" = 3 k [eu].': 1 ka 0.05 3 K= 0.05 7 2) Marginal distribution 3 3 S 618,4) Skedua に 0.05 [64] du a o 3 0.05 [eu] cosce-e'] -0.95

3) Since the probability can't be negative the answer will be zero

correlation coefficient is given by COM Cox (x, y) V var (x) varty) variance = Etx?] - ECX] 2 fazloga da - ( set (x) de) 2 var (x) = 22 ore doc - (S el dx) Je 11ly Eły?? -ELYI cou be,y[x,y) = xy) - ELECY] dinally correlation coefficients can be obtained