Question

Let X ~ N(1,3) and Y~ N(5,7) be two independent random variables. Find...

Var(X + Y + 32)

Var(X -Y)

Var(2X - 4Y)

Answer #1

Let X and Y be two independent random variables. Given the
marginal pdfs indicated below, find the cdf of Y/X. (Hint: Consider
two cases, 0 ≤ w ≤ 1 and 1.) (a) fx (x) =1, 0 ≤ x ≤ 1, and fγ
(y)=1, 0 ≤ y ≤ 1 (b) fx (x)=2x,0 ≤x ≤1, and fy(y)=2y, 0 ≤y ≤1

Let X and Y be independent and identically
distributed random variables with mean μ and variance
σ2. Find the following:
a) E[(X + 2)2]
b) Var(3X + 4)
c) E[(X - Y)2]
d) Cov{(X + Y), (X - Y)}

If X, Y are random variables with E(X) = 2, Var(X) = 3, E(Y) =
1, Var(Y) =2, ρX,Y = −0.5
(a) For Z = 3X − 1 find µZ, σZ.
(b) For T = 2X + Y find µT , σT
(c) U = X^3 find approximate values of µU , σU

Let X and Y be two independent random variables with ??=3, ??=2,
??=6, and ??=1.
Find ?(5?−3?+2)−?(8?−3?+7).
Your answer should be a whole number.

Let X and Y be two random variables which follow standard normal
distribution. Let J = X − Y . Find the distribution function of J.
Also find E[J] and Var[J].

7.
Let X and Y be two independent and identically distributed
random variables with expected value 1 and variance 2.56.
(i) Find a non-trivial upper bound for
P(| X + Y -2 | >= 1)
(ii) Now suppose that X and Y are independent and identically
distributed N(1;2.56) random variables. What is P(|X+Y=2| >= 1)
exactly? Briefly, state your reasoning.
(iii) Why is the upper bound you obtained in Part (i) so
different from the exact probability you obtained in...

Let T1 and T2 be
independent random variables with Var(T1) = 9
and Var(T2) = 3. Compute
Var(T1 –T2).

Let X and Y be independent random variables, uniformly
distribued on the interval [0, 2]. Find E[e^(X+Y) ].

Let X, Y be two random variables with a joint pmf
f(x,y)=(x+y)/12 x=1,2 and y=1,2
zero elsewhere
a)Are X and Y discrete or continuous random variables?
b)Construct and joint probability distribution table by writing
these probabilities in a rectangular array, recording each marginal
pmf in the "margins"
c)Determine if X and Y are Independent variables
d)Find P(X>Y)
e)Compute E(X), E(Y), E(X^2) and E(XY)
f)Compute var(X)
g) Compute cov(X,Y)

Given that Var(X) = 5 and Var(Y) = 3, and Z is defined as Z =
-2X + 4Y - 3.
(a) Find the variance of Z if X and Y are independent.
(b) If Cov (X,Y) = 1, find the variance of Z.
(c) If Cov (X,Y) = 1, compute the correlation of X and Y.

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