Question

Suppose that X ∼ Unif[0, 3] and Y is independent of X and exponentially distributed with rate 2.

Find the pdf of

(a) max{X,Y}. (b) min{X,Y}.

Answer #1

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

Let X i ~ Unif(0, 1) for 1 <= i <= n be IID (independent
identically distributed) random variables. Let Y = max(X 1 , …, X n
). What is E(Y)?

Suppose that X is uniformly distributed on the interval [0,5], Y
is uniformly distributed on the interval [0,5], and Z is uniformly
distributed on the interval [0,5] and that they are
independent.
a)find the expected value of the max(X,Y,Z)
b)what is the expected value of the max of n independent random
variables that are uniformly distributed on [0,5]?
c)find pr[min(X,Y,Z)<3]

Let X ∼ UNIF(0, 1). Find the pdf of Y = 1 − X using the
distribution-function technique. Also indicate the support of
Y.

Let X ∼ UNIF(0, 1). Find the pdf of Y = −5 ln(X) using the
transformation technique. Note that Y is an exponential random
variable. What is its parameter? Show your work.

For independent X and Y, we have probability density function
for them where pdf of X is f(x) = ne^-nx and pdf of Y is f(y) =
me^-my. (x and y greater than 0). Let M1=max(X,Y) and M2=min(X,Y).
Find cov(M2,M1).

For independent X and Y, we have probability density function
for them where pdf of X is f(x) = ne^-nx and pdf of Y is f(y) =
me^-my. (x and y greater than 0). Let M1=max(X,Y) and M2=min(X,Y).
Find cov(M2,M1).

For independent X and Y, we have probability density function
for them where pdf of X is f(x) = ne^-nx and pdf of Y is f(y) =
me^-my. (x and y greater than 0). Let M1=max(X,Y) and M2=min(X,Y).
Find cov(M2,M1).

Suppose that X is uniformly distributed on the interval [0,10],
Y is uniformly distributed on the interval [0,10], and Z is
uniformly distributed on the interval [0,10] and that they are
mutually independent.
a)find the expected value of the min(X,Y,Z)
b)find the standard deviation of the min(X,Y,Z)

Let X and Y be independent exponentially distributed stochastic
variables with parameters α and β. Find the distribution function
(c.d.f.) of X / Y.
Please show work involved and general equations used. As much
supplementary text as possible will be greatly appreciated

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