Question

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)

Answer #1

Here,

a)

We know,

So,

On comparing both sides, we get,

So,

Also,

b)

We know,

So,

On comparing both sides, we get,

So,

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).

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 descrete random variable X~Poisson(6).
Find:
Probability P(X=5)
Probability P(X=2)
Probability P(X<3)
Probability P(X>6)
μX
σX

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).

A random variable X has its probability function given by
x
0
1
2
3
4
f(x)
0.3c
0.1c
c
0.2c
0.4c
a) Find c and F(x), the cumulative distribution function for X
(for all real values of X).
b) Find the probabilities of the event X = 6 and the event X
>= 4.
c) Find P(1 < X <= 4) and P(1 < X <= 4 | X <=
3).

Let X be a random variable with density function f(x) = 2 5 x
for x ∈ [2, 3] and f(x) = 0, otherwise. (a) (6 pts) Compute E[(X −
2)3 ] without attempting to find the density function of Y = (X −
2)3 . (b) (6 pts) Find the density function of Y = (X − 2)3

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)

Let X be a normally distributed random variable with mean 5.
1. If P(4 ≤ X ≤ 4.6) = .240, find P(5.4 ≤ X ≤ 6).
2. Suppose P(X ≤ k) = .5 for some number k. What is the value of
k?
3. Suppose P(X ≤ ℓ.) = .841, for some mystery value
ℓ.. Approximately how many standard deviations above or
below the mean is the value ℓ.?

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 ).

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...

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