Question

If X, Y, and Z are independent and identically distributed Γ(1,1), derive the joint distribution of U = X+Y, V = X + Z, and W = Y + Z.

Answer #1

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Let X and Y independent random variables with U distribution (−1,1). Using the Jacobian method, determine the joint distribution of Z=X-Y and W= X+Y.

Let X and Y be independent and identically distributed
with an exponential distribution with parameter 1, Exp(1).
(a) Find the p.d.f. of Z = Y/X.
(b) Find the p.d.f. of Z = X − Y .

Continuous random variables X1 and X2 with joint density
fX,Y(x,y) are independent and identically distributed with expected
value μ.
Prove that E[X1+X2] = E[X1] +E[X2].

Let X and Y be independent N(0,1) RVs. Suppose U =
(X+Y)/squareroot(2) and V = (X-Y)/squareroot(2).
Please derive the joint distribution of (U, V ) by using the
Jacobian matrix method.

Let X be exponentially distributed with paramter 2, and let Y be
exponentially distributed with parameter 4. Suppose X and Y are
independent.
(a) Let Z = Y/X. Determine the cdf and pdf of Z. (b) Deﬁne two
random variables V and W by V = X + Y, W = X −Y Determine the joint
pdf of V and W, and sketch the region in the vw-plane on which the
joint pdf is nonzero

X and Y are independent and identically distributed variables
uniform over [0,1]. Find PDF of A=Y/X

Let X∼Γ(a1,b) be independent of Y, and suppose W=X+Y∼Γ(a2,b),
where a2> a1. ShowY∼Γ(a2−a1,b)

X is uniformly distributed on the interval (0, 1), and Y is
uniformly distributed on the interval (0, 2). X and Y are
independent. U = XY and V = X/Y .
Find the joint and marginal densities for U and V .

(9) Let X and Y be iid Exp(1) RV’s. Define U = X / (X+Y) and V =
X + Y . Show your Work.
(a) Derive the joint density for (U, V ).
(b) What is the marginal distribution for U?
(c) Find the conditional mean E(X | V = 2).
(d) Are U and V independent? Explain why

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...

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