Question

a) let X follow the probability density function f(x):=e^(-x) if x>0, if Y is an independent random variable following an identical distribution f(x):=e^(-x) if x>0, calculate the moment generating function of 2X+3Y

b) If X follows a bernoulli(0.5), and Y follows a Binomial(3,0.5), and if X and Y are independent, calculate the probability P(X+Y=3) and P(X=0|X+Y=3)

Answer #1

Let the probability density function of the random variable X be
f(x) = { e ^2x if x ≤ 0 ;1 /x ^2 if x ≥ 2 ; 0 otherwise}
Find the cumulative distribution function (cdf) of X.

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 ).

1)Calculate the MGF of X^2+Y, if X follows a Bernoulli(0.5), and
Y follows a Binomial(3,0.5), and if X and Y are independent.
2) Let X follow PDF f(x):= exp(-x^2/2)/√(2π),
for -∞<x<∞. The corresponding Cumulative Distribution
Function is denoted by F(x)=P(X<=x). If Y is independent with X,
follows the same distribution as X, what is the probability that
the minimum of X and Y will be positive.

Let X have the distribution that has the following probability
density function:
f(x)={2x,0<x<1
{0, Otherwise
Find the probability that X>0.5.
Why is the probability 0.75 and not 0.5?

let X follow the probability density function f(x):=e^(-x) if
x>0.
For what value of k is the probability that X is greater than k
is at least 0.75

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)=?

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)

X and Y are two random variables that follow the probability
density function of f ( x , y ) = c x 2 y for 0<x<3 and
0<y<2.
Determine the following:
(a) P(X<1&Y<1)
(b) P(1<Y<2.5)
(c) COV(X,Y)
(d) Conditional probability distribution of Y given that
X=1.

Independent random variables X and Y follow binomial
distributions with parameters(n1,θ) and (n2,θ). Let Z =X+Y. What
will be the distribution of Z?
Hint: Use moment generating function.

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.

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