Question

The proportion of impurities per batch in a chemical product is a random variable Y with density function

f(y) = 12y2(1 − y), 0 ≤ y ≤ 1,

0, elsewhere

. Find the mean and variance of the percentage of impurities in a randomly selected batch of the chemical.

E(Y) = ??:

. V(Y) =??

Answer #1

The percentage of impurities per batch in a chemical product is
a random variable X with density function
12x2(1−x), 0≤x≤1 f(x) =
0, elsewhere A batch with more than 40% impurities cannot be
sold.
(a) What is the distribution of X?

For certain ore samples, the proportion Y of impurities
per sample is a random variable with density function
f(y) =
5
2
y4 +
y,
0 ≤ y ≤ 1,
0,
elsewhere.
The dollar value of each sample is
W = 8 − 0.8Y.
Find the mean and variance of W. (Round your answers to
four decimal places.)
E(W) = V(W) =

The proportion of impurities in certain ore samples is a random
variable Y with a density function given by
f(y) =
3
2
y2 + y,
0 ≤ y ≤ 1,
0,
elsewhere.
The dollar value of such samples is U = 3 − Y/8
. Find the probability density function for U
.

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 ).

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...

Let Y have density function
f(y) =
cye−9y,
0 ≤
y ≤ ∞,
0,
elsewhere.
b) Give the mean and variance for Y.
E(Y)=
V(Y)=
(c) Give the moment-generating function for Y.
m(t) =___________, t<9

let X be a random variable that denotes the life (or time to
failure) in hours of a certain electronic device. Its probability
density function is given by
f(x){ 0.1 e−0.1x, x > 0 , 0 , elsewhere
(a) What is the mean lifetime of this type of device?
(b) Find the variance of the lifetime of this device.
(c) Find the expected value of X2 − 20X + 100.

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.

The density
function for the random variable Y is ?(?) = 64(? +
2)−5
for y>0. Find
the variance 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)

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