Question

Let X_{1}, X_{2} , X_{3} be independent
random variables that represent lifetimes (in hours) of three key
components of a device. Say their respective distributions are
exponential with means 1000, 1500, and 1800. Let Y be the minimum
of X_{1}, X_{2}, X_{3} and compute P(Y >
1000).

Answer #1

Let X1, X2, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is
the middle of the three values). Find the conditional CDF of X1,
given the event Y = 1/2. Under this conditional distribution, is X1
continuous? Discrete?

Let X1, X2, X3, and X4 be mutually independent random variables
from the same distribution. Let
S = X1 + X2 + X3 + X4. Suppose we know that S is a Chi-Square
random variable with 2 degrees of freedom. What
is the distribution of each of the Xi?

Let X1 and X2 be independent random variables such that X1 ∼ P
oisson(λ1) and X2 ∼ P oisson(λ2). Find the distribution of Y = X1 +
X2.s

Let X1 and X2 be independent Poisson
random variables with respective parameters λ1 and
λ2. Find the conditional probability mass function
P(X1 = k | X1 + X2 = n).

Let X1,X2,..., Xn be independent random variables that are
exponentially distributed with respective parameters λ1,λ2,...,
λn.
Identify the distribution of the minimum V =
min{X1,X2,...,Xn}.

Let
x1, x2 x3 ....be a sequence of independent and identically
distributed random variables, each having finite mean E[xi] and
variance Var（xi）.
a）calculate the var （x1+x2）
b）calculate the var（E[xi]）
c） if n-> infinite, what is Var（E[xi]）？

Suppose X1, X2, X3, and
X4 are independent and identically distributed random
variables with mean 10 and variance 16. in addition, Suppose that
Y1, Y2, Y3, Y4, and
Y5are independent and identically distributed random
variables with mean 15 and variance 25. Suppose further that
X1, X2, X3, and X4 and
Y1, Y2, Y3, Y4, and
Y5are independent. Find Cov[bar{X} + bar{Y} + 10,
2bar{X} - bar{Y}], where bar{X} is the sample mean of
X1, X2, X3, and X4 and
bar{Y}...

If X1 and X2 are independent exponential random variables with
respective parameters λ1 and λ2, find the distribution of Z =
min{X1, X2}.

If X1 and X2 are independent exponential random variables with
respective parameters 1 and 2, find the distribution of Z = min{X1,
X2}.

Suppose that X1,X2 and X3 are independent random variables with
common mean E(Xi) = μ and variance Var(Xi) = σ2. Let V= X2−X3 and W
= X1− 2X2 + X3.
(a) Find E(V) and E(W).
(b) Find Var(V) and Var(W).
(c) Find Cov(V,W).
(d) Find the correlation coefficient ρ(V,W). Are V and W
independent?

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