Question

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)

Answer #1

A) 0.9545

B) 0.6827

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)

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 be two independent random variables with
μX =E(X)=2,σX =SD(X)=1,μY =2,σY =SD(Y)=3.
Find the mean and variance of
(i) 3X
(ii) 6Y
(iii) X − Y

You have two random variables X and Y
X -> μX = 5 , σX = 3
Y -> μY = 7 , σY = 4
Now, we define two new random variables
Z = X - Y
W = X + Y
Answer the below questions:
μZ =
[ Select ]
["3",
"1", "-2"]
σZ =
...

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

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)

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

Let descrete random variable X~Poisson(6).
Find:
Probability P(X=5)
Probability P(X=2)
Probability P(X<3)
Probability P(X>6)
μX
σX

Let (X1, Y1), . . . ,(Xn, Yn), be a random sample from a
bivariate normal distribution with parameters µ1, µ2, σ2 1 , σ2 2 ,
ρ. (Note: (X1, Y1), . . . ,(Xn, Yn) are independent). What is the
joint distribution of (X ¯ , Y¯ )?

Let the random variable X and Y have the joint pmf f(x, y) =
xy^2/c where x = 1, 2, 3; y = 1, 2, x + y ≤ 4 , that is, (x, y) are
{(1, 1),(1, 2),(2, 1),(2, 2),(3, 1)} .
(a) Find c > 0 .
(b) Find μX
(c) Find μY
(d) Find σ^2 X
(e) Find σ^2 Y
(f) Find Cov (X, Y )
(g) Find ρ , Corr (X, Y )
(h) Are X...

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