Question

Let the random variable X and Y have the joint pmf f(x, y) = xy^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 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?

Answer #1

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 continuous random variables with joint
distribution function F(x, y), and let g(X, Y ) and h(X, Y ) be
functions of X and Y . Prove the following:
(a) E[cg(X, Y )] = cE[g(X, Y )].
(b) E[g(X, Y ) + h(X, Y )] = E[g(X, Y )] + E[h(X, Y )].
(c) V ar(a + X) = V ar(X).
(d) V ar(aX) = a 2V ar(X).
(e) V ar(aX + bY ) = a...

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 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

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

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.

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

Let X and Y be continuous random variable with joint pdf
f(x,y) = y/144 if 0 < 4x < y < 12 and
0 otherwise
Find Cov (X,Y).

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

Consider two random variable X and Y with joint PMF given in the
table below.
Y = 2
Y = 4
Y = 5
X = 1
k/3
k/6
k/6
X = 2
2k/3
k/3
k/2
X = 3
k
k/2
k/3
a) Find the value of k so that this is a valid PMF. Show your
work.
b) Re-write the table with the joint probabilities using the
value of k that you found in (a).
c) Find the marginal...

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