Question

Consider the density function:

?(?) = {

6?(1 − ?) 0 < ? < 1 0 ?????ℎ???

i) Find ? and ?. ii) Compute ?(? − ? < ? < ? + ?)

Answer #1

TOPIC:Mean,variance,sd and probability.

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)=?

Consider the joint density function f (x, y) = 1 if 0<=
x<= 1; 0<=y<= 1. [0 elsewhere]
a) Obtain the probability density function of the v.a Z, where Z =
X^2.
b) Obtain the probability density function of v.a W, where W =
X*Y^2.
c) Obtain the joint density function of Z and W, that is, g (Z,
W)

Let X and Y be two continuous random variables with joint
probability density function
?(?, ?) = { ? 2 + ?? 3 0 ≤ ? ≤ 1, 0 ≤ ? ≤ 2 0 ??ℎ??????
Find ?(? + ? ≥ 1). Sketch the surface in the ? − ? plane.

Compound probability density function of random variables ? and
?
??, ? (?, ?) = (6/7 (? * y + (? * ?) / 2), 0 <? <2, 0 <?
<1?, other
It is given as. According to this
(a) Show that this function is a probability density
function.
(b) Find the marginal probability density function of ?.
(c) Calculate the probability ? {?> ?}.
(d) Find the expected value of ? [?].

consider the joint density function
Fx,y,za (x,y,z)=(x+y)e^(-z) where 0<x<1, 0<y<1,
z>0
find the marginal density of z : fz (z).
hint. figure out which common distribution Z follows and report
the rate parameter
integral (x+y)e^(-z) dz
(x+y)(-e^(-z) + C
is my answer 1. ???

1. Consider the functions ?(?) = √? + 1 , ?(?) = 2? 4−? , and
?(?) = ? 2 − 5
(a) Find ?(0), ?(0), ?(0)
(b) (??)(?)
(c) (? ∘ ?)(?)
(d) Find the domain of (? ∘ ?)(?)
(e) Find and simplify ?(?+ℎ)−?(?) ℎ .
(f) Determine if ? is an even function, odd function or neither.
Show your work to justify your answer.
2. Sketch the piecewise function. ?(?) = { |? + 2|, ??? ?...

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....

Consider the probability density function f(x) = (3θ +
1) x3θ, 0 ≤
x ≤ 1.
The random sample is 0.859, 0.008, 0.976, 0.136, 0.864, 0.449,
0.249, 0.764. The moment estimator of θ based on a random
sample of size n is .055. Please answer the following:
a) find the maximum likelihood estimator of θ based on
a random sample of size n. Then use your result to find
the maximum likelihood estimate of θ based on the given
random...

Let X be a random variable with probability density function fX
(x) = I (0, 1) (x). Determine the probability density function of Y
= 3X + 1 and the density function of probability of Z = - log
(X).

LetX1 andX2 have joint density function f(x1,x2) =( 30x1x2^2, x1
−1≤ x2 ≤1−x1, 0≤ x1 ≤1, 0 otherwise.
(a) Find the marginal density of X1.
(b) Find the marginal density of X2.
(c) Are X1 and X2 independent?(why/why not)
(d) Find the conditional density of X2 given X1 = x1
(e) Compute Cov(X1,X2)

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