Question

A. (i) Consider the random variable X with pmf: pX (−1) = pX (1) = 1/8, pX (0) = 3/4.

Show that the Chebyshev inequality P (|X − μ| ≥ 2σ) ≤ 1/4 is actually an equation for

this random variable.

(ii) Find the pmf of a different random variable Y that also takes
the values {−1, 0, 1}

for which the Chebyshev inequality P (|X − μ| ≥ 3σ) ≤ 1/9 is actually an equation.

Answer #1

Consider the family of distributions with pmf pX(x) = p if x =
−1, 2p if x = 0, 1 − 3p if x = 1 . Here p is an unknown parameter,
and 0 ≤ p ≤ 1/3. Let X1, X2, . . . , Xn be iid with common pmf a
member of this family. Consider the statistics A = the number of i
with Xi = −1, B = the number of i with Xi = 0,...

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...

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

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 X be a random variable with the pmf p(x) which is
positive at x=1;0;1, and zero elsewhere. If E(X^3) = 0 andE(X^2)
=p(0),what is p(1)?

Consider a random variable X such that:
??(?) =|?| / 2? ??? ? ∈ {−2, −1, 1, 2},
??(?) = 0 ??? ? ∉ {−2, −1, 1, 2},
Where ? > 0 is a real parameter.
a) Find a.
b) What is the PMF of the random variable ? =?^2+1/??
Guidance: ? is a function of ? (? = ?(?)). Write P(? = ?) in terms
of
P(? = ?) such that ? = ?(?). You can make use of...

Let f(x)=c(4/9)^x. Find c such that f represents the pmf of a
random variable whose possible values are 1, 2, 3, ... .
A.4/5
B.5/4
C.4/9
The above random variable is:
A.Geom(4/5) random variable
B.Bern(4/9) random variableC
c.Geom(5/9) random variable

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

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

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