Question

1. A and b are not correlated, treat as separate problems

a) The random variable X has uniform continuous distribution on
the interval [0, 10]. **Find the distribution of Y = X3 and
P(Y > 50).**

b) The random variables X and Y are jointly bivariate normal with parameters µX = 0, σX = 1, µY = 0, σY = 2 and ρ = 0.9.

**i) Find P(Y > 0)**

**ii) Find P(Y > 0|X = 1)**

Answer #1

Exercise 10.45. Suppose that the joint distribution of X,Y is
bivariate normal with parameters σX,σY,ρ,µX,µY as described in
Section 8.5. (a) Compute the conditional probability density of X
given Y =y. (b) Find E[X|Y].

Let X and Y have a bivariate normal distribution with parameters
μX = 0, σX = 3; μY = 8,
σY = 5; ρ = 0.6. Find the following probabilities.
P(-6 < X < 6)
P(6 < Y < 14 | X = 2)

Let X and Y have a bivariate normal distribution with parameters
μX = 0, σX = 3; μY = 8, σY = 5; ρ = 0.6. Find the following
probabilities.
(A) P(-6 < X < 6)
(B) P(6 < Y < 14 | X = 2)

. [10] Profit in thousands from a project is determined by the
equation Profit = 2X + 3Y where X and Y are random variables that
can be modeled by a normal distribution. For this problem µx is 20,
µy is 30, σx is 2 and σy is 3. Find the mean and standard deviation
of Profit in thousands.

Given a random variable X following normal distribution with
mean of -3 and standard deviation of 4. Then random variable
Y=0.4X+5 is also normal.
(1)Find the distribution of Y, i.e. μy,σy
(2)Find the probabilities P(−4<X<0),P(−1<Y<0)
(3)Find the probabilities(let n size =8)
P(−4<X¯<0),P(3<Y¯<4)
(4)Find the 53th percentile of the distribution of X

a) Suppose that X is a uniform continuous random variable where
0 < x < 5. Find the pdf f(x) and use it to find P(2 < x
< 3.5).
b) Suppose that Y has an exponential distribution with mean 20.
Find the pdf f(y) and use it to compute P(18 < Y < 23).
c) Let X be a beta random variable a = 2 and b = 3. Find P(0.25
< X < 0.50)

Problems 9 and 10 refer to the discrete random variables X and Y
whose joint distribution is given in the following table.
Y=-1
Y=0
Y=1
X=1
1/4
1/8
0
X=2
1/16
1/16
1/8
X=3
1/16
1/16
1/4
P9: Compute the marginal distributions of X and Y, and use these
to compute E(X), E(Y), Var(X), and Var(Y).
P10: Compute Cov(X, Y) and the correlation ρ for the random
variables X and Y. Are X and Y independent?

Consider the following bivariate distribution p(x, y) of two
discrete random variables X and Y.
Y\X
-2
-1
0
1
2
0
0.01
0.02
0.03
0.10
0.10
1
0.05
0.10
0.05
0.07
0.20
2
0.10
0.05
0.03
0.05
0.04
a) Compute the marginal distributions p(x) and p(y)
b) The conditional distributions P(X = x | Y = 1)
c) Are these random variables independent?
d) Find E[XY]
e) Find Cov(X, Y) and Corr(X, Y)

Calculate the quantity of interest please.
a) Let X,Y be jointly continuous random variables generated as
follows: Select X = x as a uniform random variable on [0,1]. Then,
select Y as a Gaussian random variable with mean x and variance 1.
Compute E[Y ].
b) Let X,Y be jointly Gaussian, with mean E[X] = E[Y ] = 0,
variances V ar[X] = 1,V ar[Y ] = 1 and covariance Cov[X,Y ] = 0.4.
Compute E[(X + 2Y )2].

Let X and Y be independent random variables with means EX = 10
and EY = 5 and standard deviations σX = 2 and
σY = 1.
Find the second moment E(X + Y + 1)2

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