Question

Y is a continuous random variable with a probability density function f(y)=a+by and 0<y<1. Given E(Y^2)=1/6, Find:

i) a and b.

ii) the moment generating function of Y. M(t)=?

Answer #1

10pts) Let Y be a continuous random variable with density
function f(y) = 1 2 e −|y| , −∞ < y < ∞ 0, elsewhere (a) Find
the moment-generating function of Y . (b) Use the moment-generating
function you find in (a) to find the V (Y ).

Let X be a continuous random variable with the following
probability density function:
f(x) = e^−(x−1) for x ≥ 1; 0 elsewhere
(i) Find P(0.5 < X < 2).
(ii) Find the value such that random variable X exceeds it 50%
of the time. This value is called the median of the random variable
X.

(i) If a discrete random variable X has a moment generating
function
MX(t) = (1/2+(e^-t+e^t)/4)^2, all t
Find the probability mass function of X. (ii) Let X and Y be two
independent continuous random variables with moment generating
functions
MX(t)=1/sqrt(1-t) and MY(t)=1/(1-t)^3/2, t<1
Calculate E(X+Y)^2

Let X be a random variable with probability density function
given by
f(x) = 2(1 − x), 0 ≤ x ≤ 1,
0, elsewhere.
(a) Find the density function of Y = 1 − 2X, and find E[Y ] and
Var[Y ] by using the derived density function. (b) Find E[Y ] and
Var[Y ] by the properties of the expectation and the varianc

A continuous random variable X has the following
probability density function F(x) = cx^3, 0<x<2 and 0
otherwise
(a) Find the value c such that f(x) is indeed
a density function.
(b) Write out the cumulative distribution function of
X.
(c) P(1 < X < 3) =?
(d) Write out the mean and variance of X.
(e) Let Y be another continuous random variable such
that when 0 < X < 2, and 0 otherwise. Calculate
the mean of Y.

Let Y be a random variable with a given probability density
function by f (y) = y + ay ^ 2, with y E [0; 1] and a E [0; 2].
Determine: The value of a.
The Y distribution function.
The value of P (0,5 < Y < 1)

Probability density function of the continuous random variable X
is given by f(x) = ( ce −1 8 x for x ≥ 0 0 elsewhere
(a) Determine the value of the constant c.
(b) Find P(X ≤ 36).
(c) Determine k such that P(X > k) = e −2 .

The probability density function for a continuous random
variable X is given by
f(x) =
0.6 0<X<1
=
0.10(x) 1 ≤X≤ 3
=
0 otherwise
Find the 85th percentile value of X.

3. (10pts) Let Y be a continuous random variable having a gamma
probability distribution with expected value 3/2 and variance 3/4.
If you run an experiment that generates one-hundred values of Y ,
how many of these values would you expect to find in the interval
[1, 5/2]?
4. (10pts) Let Y be a continuous random variable with density
function f(y) = 1 2 e −|y| , −∞ < y < ∞ 0, elsewhere (a) Find
the moment-generating function of...

6. A continuous random variable X has probability density
function
f(x) =
0 if x< 0
x/4 if 0 < or = x< 2
1/2 if 2 < or = x< 3
0 if x> or = 3
(a) Find P(X<1)
(b) Find P(X<2.5)
(c) Find the cumulative distribution function F(x) = P(X< or
= x). Be sure to define the function for all real numbers x. (Hint:
The cdf will involve four pieces, depending on an interval/range
for x....

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