Question

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)?

Answer #1

Let X be a discrete random variable with the pmf
p(x): 0.8 for x=-4,
0.1 for x=-2,
0.07 for x=0,
0.03 for x=2
a) Find E(2/X)
b) Find E(lXl)
c) Find Var(lXl)

Let X be a discrete random variable with probability mass
function (pmf) P (X = k) = C *ln(k) for k = e; e^2 ; e^3 ; e^4 ,
and C > 0 is a constant.
(a) Find C.
(b) Find E(ln X).
(c) Find Var(ln X).

Given a random variable X has the following pmf:
X
-1
0
1
P[X]
0.25
0.5
0.25
Define Y = X2 & W= Y+2.
Which one of the following statements is not true?
A) V[Y] = 0.25.
B) E[XY] = 0.
C) E[X3] = 0.
D) E[X+2] = 2.
E) E[Y+2] = 2.5.
F) E[W+2] = 4.5.
G) V[X+2] = 0.5.
H) V[W+2] = 0.25.
I) P[W=1] = 0.5
J) X and W are not independent.

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)

1. Let X be a discrete random variable with the probability mass
function P(x) = kx2 for x = 2, 3, 4, 6.
(a) Find the appropriate value of k.
(b) Find P(3), F(3), P(4.2), and F(4.2).
(c) Sketch the graphs of the pmf P(x) and of the cdf F(x).
(d) Find the mean µ and the variance σ 2 of X. [Note: For a
random variable, by definition its mean is the same as its
expectation, µ = E(X).]

Let f(x) = (1/2)^x, x = 1,2,3,... be the PMF of the random
variable X. Find the MGF, mean, and variance of X.

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.

Let X be a r.v with pmf p(x) = c( 2 /3 )^ x , x = 0, 1, 2, 3,
... (infinitely many values of x)
(a) Find the constant c. (b) With the constant you find in (a),
find the mean E(X)

Let N be a positive integer random variable with PMF of the form
pN(n)=12⋅n⋅2−n,n=1,2,…. Once we see the numerical value of N , we
then draw a random variable K whose (conditional) PMF is uniform on
the set {1,2,…,2n} . 1. Find joint PMF pN,K(n,k) For n=1,2,… and
k=1,2,…,2n 2. Find the marginal PMF pK(k) as a function of k . For
simplicity, provide the answer only for the case when k is an even
number. For k=2,4,6,… 3. Let...

let x be a discrete random variable with positive integer
outputs.
show that P(x=k) = P( x> k-1)- P( X>k) for any positive
integer k.
assume that for all k>=1 we have P(x>k)=q^k. use (a) to
show that x is a geometric random variable.

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