Question

In: Statistics and Probability

Let the random variable X and Y have the joint pmf f(x, y) = , c...

Let the random variable X and Y have the joint pmf f(x, y) = , c xy 2 where x = 1, 2, 3; y = 1, 2, x + y ≤ 4 , that is, (x, y) are {(1, 1),(1, 2),(2, 1),(2, 2),(3, 1)} .

(a) Find c > 0 .

(b) Find μ . X

(d) Find σ . 2 X

(e) Find σ . 2 Y

(f) Find Cov (X, Y ) .

(g) Find ρ , Corr (X, Y ) .

(h) Are X and Y independent?

Please show work/explanation if you can, thank you!

Solutions

Expert Solution

We would be looking at the first 4 parts of the question here as:

a) The value of C can be computed by summing up the probability values on X, Y range as 1.

P(1, 1) = c,
P(1, 2) = 4c,
P(2, 1) = 2c,
P(2,2) = 8c,
P(3,1) = 3c

Total probability = 18c = 1

c = 1/18 is the required value of c here.

b) The marginal PDF for X is first obtained here as:
P(X = 1) = P(1, 1) + P(1,2) = c + 4c = 5c = 5/18
P(X = 2) = P(2, 1) + P(2, 2) = 10c = 10/18
P(X = 3) = 3/18

Therefore the mean of X here is obtained as:

= E(X) = 1*(5/18) + 2*(10/18) + 3*(3/18) = 17/9

Therefore 17/9 = 1.2143 is the required mean value of X here.

c) Now similar to the above method, we obtain the marginal PDF for Y here as:
P(Y = 1) = c + 2c + 3c = 6c = 1/3
P(Y = 2) = 12c = 2/3

E(Y) = 1*(1/3) + 2*(2/3) = 5/3

Therefore 5/3 = 1.6667 is the required mean of Y here.

d) The second moment of X here is obtained as:
E(X²) = 1²*(5/18) + 2²*(10/18) + 3²*(3/18) = 4

Therefore, the variance now is computed here as:

= E(X²) - [E(X)]² = 4 - (17/9)² = 0.4321

therefore 0.4321 is the required variance of X here.

orchestra answered 1 year ago

Next > < Previous

Related Solutions

Let the random variable and have the joint pmf X Y f(x,y) = {x(y)^2}/c where x...

Let the random variable and have the joint pmf X Y f(x,y) = {x(y)^2}/c where x = 1, 2, 3 ; y = 1, 2, x+y<= 4, that is (x,y) are {(1,1), (1,2), (2,1), (2,2), 3,1)} (a) Find . c > 0 (b) Find . μX (c) Find . μY (d) Find . σ2 X (e) Find . σ2 Y (f) Find Cov . (X,Y ) (g) Find p , Corr . (x,y) (h) Are and X and Y independent

Let the joint pmf of X and Y be defined by f (x, y) = c(x...

Let the joint pmf of X and Y be defined by f (x, y) = c(x + y), x =0, 1, 2, y = 0, 1, with y ≤ x. 1. Are X and Y independent or dependent? Why or why not? 2. Find g(x | y) and draw a figure depicting the conditional pmfs for y =0 and 1. 3. Find h(y | x) and draw a figure depicting the conditional pmfs for x = 0, 1 and2. 4....

Let the joint pmf of X and Y be defined by f (x, y) = c(x...

Let the joint pmf of X and Y be defined by f (x, y) = c(x + y), x =0, 1, 2, y = 0, 1, with y ≤ x. 1. Find g(x | y) and draw a figure depicting the conditional pmfs for y =0 and 1. 2. Find h(y | x) and draw a figure depicting the conditional pmfs for x = 0, 1 and2. 3. Find P(0 < X <2 |Y = 0), P(X ≤ 2 |Y...

let p ( x , y )be a joint pmf of Xand Y. p ( x...

let p ( x , y )be a joint pmf of Xand Y. p ( x , y ) y = 0 y = 1 y = 2 x = 0 .12 .10 .08 x = 1 .13 .17 .10 x = 2 .15 .15 0 (a) Find the marginal pmf's of Xand Y. (b) Determine whether Xand Yare independent. c) Find Correlation (X,Y)

SOLUTION REQUIRED WITH COMPLETE STEPS Let X and Y be discrete random variables, their joint pmf...

SOLUTION REQUIRED WITH COMPLETE STEPS Let X and Y be discrete random variables, their joint pmf is given as Px,y = ?(? + ? + 2)/(B + 2) for 0 ≤ X < 3, 0 ≤ Y < 3 (Where B=0) a) Find the value of ? b) Find the marginal pmf of ? and ? c) Find conditional pmf of ? given ? = 2

let the random variables x and y have the joint p.m.f f(x,y)= (x+y)/12 where x=1,2 and...

let the random variables x and y have the joint p.m.f f(x,y)= (x+y)/12 where x=1,2 and y=1,2 calculate the covariance of x and y calculate the correlation coefficient of x and y Thank you

Let X and Y have joint PDF f(x) = c(e^-(x/λ + y/μ)) 0 < x <...

Let X and Y have joint PDF f(x) = c(e^-(x/λ + y/μ)) 0 < x < infinity and 0 < y < infinity with parameters λ > 0 and μ > 0 a) Find c such that this is a PDF. b) Show that X and Y are Independent c) What is P(1 < X < 2, 0 < Y < 5) ? Leave in exponential form d) Find the marginal distribution of Y, f(y) e) Find E(Y)

let the continuous random variables X and Y have the joint pdf: f(x,y)=6x , 0<x<y<1 i)...

let the continuous random variables X and Y have the joint pdf: f(x,y)=6x , 0<x<y<1 i) find the marginal pdf of X and Y respectively, ii) the conditional pdf of Y given x, that is fY|X(y|x), iii) E(Y|x) and Corr(X,Y).

Define the joint pmf of (X, Y) by f(0, 10) = f(0, 20) = 1 /...

Define the joint pmf of (X, Y) by f(0, 10) = f(0, 20) = 1 / 24, f(1, 10) = f(1, 30) = 1 / 24, f(1, 20) = 6 / 24, f(2, 30) = 14 / 24 Find the value of the following. Give your answer to three decimal places. a) E(Y | X = 0) = b) E(Y | X = 1) = c) E(Y | X = 2) = d) E(Y) =

Let X and Y be continuous random variables with joint pdf f(x, y) = kxy^2 0...

Let X and Y be continuous random variables with joint pdf f(x, y) = kxy^2 0 < x, 0 < y, x + y < 2 and 0 otherwise 1) Find P[X ≥ 1|Y ≤ 1.5] 2) Find P[X ≥ 0.5|Y ≤ 1]

Subjects