Question

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).

Answer #1

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 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)

Suppose X is a discrete random variable with probability mass
function given by
p (1) = P (X = 1) = 0.2
p (2) = P (X = 2) = 0.1
p (3) = P (X = 3) = 0.4
p (4) = P (X = 4) = 0.3
a. Find E(X^2) .
b. Find Var (X).
c. Find E (cos (piX)).
d. Find E ((-1)^X)
e. Find Var ((-1)^X)

Q6/
Let X be a discrete random variable defined by the
following probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Give P(4≤ X < 8)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q7/
Let X be a discrete random variable defined by the following
probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Let F(x) be the CDF of X. Give F(7.5)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q8/
Let X be a discrete random variable defined by the following
probability function :
x
2
6...

Consider a discrete random variable X with probability mass
function P(X = x) = p(x) = C/3^x, x = 2, 3, 4, . . . a. Find the
value of C. b. Find the moment generating function MX(t). c. Use
your answer from a. to find the mean E[X]. d. If Y = 3X + 5, find
the moment generating function MY (t).

Let X be a continuous random variable with probability density
function (pdf) ?(?) = ??^3, 0 < ? < 2.
(a) Find the constant c.
(b) Find the cumulative distribution function (CDF) of X.
(c) Find P(X < 0.5), and P(X > 1.0).
(d) Find E(X), Var(X) and E(X5 ).

Let X be a random variable with probability mass function
P(X =1) =1/2, P(X=2)=1/3, P(X=5)=1/6
(a) Find a function g such that E[g(X)]=1/3 ln(2) + 1/6 ln(5).
You answer should give at least the values g(k) for all possible
values of k of X, but you can also specify g on a larger set if
possible.
(b) Let t be some real number. Find a function g such that
E[g(X)] =1/2 e^t + 2/3 e^(2t) + 5/6 e^(5t)

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?

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);

Suppose that the probability mass function for a discrete random
variable X is given by p(x) = c x, x = 1, 2, ... , 9. Find the
value of the cdf (cumulative distribution function) F(x) for 7 ≤ x
< 8.

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