Question

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 PMFs of X and Y. Show your work.

d) Find P(X ≤ 2, Y ≤ 4)

e) Find P(Y=2 | X=1). Show your work.

f) Are X and Y independent? Justify with calculations.

Answer #1

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.

If X and Y are discrete random variables with joint PMF
P(X,Y )(x, y) = c(2x+y)(x! y!) for x = 0,
1, 2, … and y = 0, 1, 2, … and zero otherwise
a) Find the constant c.
b) Find the marginal PMFs of X and Y. Identify their
distribution along with their parameters.
c) Are X and Y independent? Why/why not?

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, 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 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

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

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?

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

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 K>0 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?
Hint:££Px,y(x,y)=1,Px(x)=£Px,y(x,y);Py(y)=£Px,y(.
x,y);E[]=£xpx(x);

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...

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