Question

The random variables, X and Y , have the joint pmf f(x,y)=c(x+2y), x=1,2 y=1,2 and zero otherwise.

1. Find the constant, c, such that f(x,y) is a valid pmf.

2. Find the marginal distributions for X and Y .

3. Find the marginal means for both random variables.

4. Find the marginal variances for both random variables.

5. Find the correlation of X and Y .

6. Are the two variables independent? Justify.

Answer #1

1) f(x, y) = c(x + 2y)

P(X = 1, Y = 1) = 3c,

P(X = 2, Y = 1) = 4c,

P(X = 1, Y = 2) = 5c,

P(X = 2, Y = 2) = 6c

Sum of all probabilities should be 1. Therefore 18c = 1

**Therefore c = 1/18**

2) The marginal distributions for X and Y are obtained here as:

P(X = 1) = 4/9 and P(X = 2) = 5/9

P(Y = 1) = 7/18 and P(Y = 2) = 11/18

3) The marginal means are computed here as:

E(X) = 1(4/9) + 2(5/9) = **14/9**

E(Y) = 1(7/18) + 2(11/18) = **29/18**

4) The marginal variances are computed as:

E(X^{2}) = 1(4/9) + 2^{2}(5/9) = 8/3

E(Y^{2}) = 1(7/18) + 2^{2}(11/18) = 17/6

Var(X) = E(X^{2}) - [E(X)]^{2} = (8/3) -
(14/9)^{2} = **20/81**

Var(Y) = E(Y^{2}) - [E(Y)]^{2} = (17/6) -
(29/18)^{2} = **77/324**

5) E(XY) = 3c + 8c + 10c + 24c = 45c = 45/18 = 15/6 = 5/2

Cov(X, Y) = E(XY) - E(X)E(Y) = (5/2) - (14/9)*(29/18) = -1/162

Therefore correlation here is computed as:

6) As cov(X,Y) is not equal to 0, **therefore the 2
variables are not independent here.**

Let X, Y be two random variables with a joint pmf
f(x,y)=(x+y)/12 x=1,2 and y=1,2
zero elsewhere
a)Are X and Y discrete or continuous random variables?
b)Construct and joint probability distribution table by writing
these probabilities in a rectangular array, recording each marginal
pmf in the "margins"
c)Determine if X and Y are Independent variables
d)Find P(X>Y)
e)Compute E(X), E(Y), E(X^2) and E(XY)
f)Compute var(X)
g) Compute cov(X,Y)

Let X and Y be discrete random variables, their joint pmf is
given as ?(x,y)= ?(? + ? − 2)/(B + 1) for 1 < X ≤ 4, 1 < Y ≤
4 Where B is the last digit of your registration number ( B=3) a)
Find the value of ? b) Find the marginal pmf of ? and ? c) Find
conditional pmf of ? given ? = 3

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=2.
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 2

Let X and Y be two continuous random variables with joint
probability density function
f(x,y) =
6x 0<y<1, 0<x<y,
0 otherwise.
a) Find the marginal density of Y .
b) Are X and Y independent?
c) Find the conditional density of X given Y = 1 /2

SOLUTION REQUIRED WITH COMPLETE STEPS
Let X and Y be discrete random variables, their joint pmf is
given as Px,y = ?(? + ?)/(B + 1) for 0 < X ≤ 3, 0 < Y ≤ 3
(Where B=5)
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 2

SOLUTION REQUIRED WITH COMPLETE STEPS
Let X and Y be discrete random variables, their joint pmf is
given as Px,y = ?(? + ? − 2)/(B + 1) for 1 < X ≤ 4, 1 < Y ≤ 4
(Where B=2)
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 3

SOLUTION REQUIRED WITH COMPLETE STEPS
Let X and Y be discrete random variables, their joint pmf is
given as Px,y = ?(? + ? + 1)/(B + 1) for 0 ≤ X < 3, 0 ≤ Y < 3
(Where B=7)
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 1

For continuous random variables X and Y with joint probability
density function. f(x,y) = xe−(x+y) when x > 0 and y
> 0 f(x,y) = 0 otherwise
a. Find the conditional density F xly (xly)
b. Find the marginal probability density function fX (x)
c. Find the marginal probability density function fY (y).
d. Explain if X and Y are independent

Find the joint discrete random variable x and y,their joint
probability mass function is given by Px,y(x,y)={k(x+y)
x=-2,0,+2,y=-1,0,+1
0 Otherwise }
2.1 determine the value of constant k,such that this will be
proper pmf?
2.2 find the marginal pmf’s,Px(x) and Py(y)?
2.3 obtain the expected values of random variables X and Y?
2.4 calculate the variances of X and Y?

Let X and Y be jointly continuous random variables with joint
density function f(x, y) = c(y^2 − x^2 )e^(−2y) , −y ≤ x ≤ y, 0
< y < ∞.
(a) Find c so that f is a density function.
(b) Find the marginal densities of X and Y .
(c) Find the expected value of X

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