Question

Let X_{1}, X_{2},... be a sequence of
independent random variables distributed exponentially with mean 1.
Suppose that N is a random variable, independent of the Xi-s, that
has a Poisson distribution with mean λ > 0. What is the expected
value of X_{1} + X_{2} +···+
X_{N}^{2}?

(A) N^{2}

(B) λ + λ^{2}

(C) λ^{2}

(D) 1/λ^{2}

Answer #1

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]）？

let X1 X2 ...Xn-1 Xn be independent exponentially distributed
variables with mean beta
a). find sampling distribution of the first order statistic
b). Is this an exponential distribution if yes why
c). If n=5 and beta=2 then find P(Y1<=3.6)
d). find the probability distribution of Y1=max(X1, X2, ...,
Xn)

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

You are given that X1 and X2 are two independent and identically
distributed random variables with a Poisson distribution with mean
2. Let Y = max{X1, X2}. Find P(Y = 1).

We have a value Y dependent on a set of independent random
variables X1, X2,..., Xn by the
following relation:
Y=X12+X22+...+Xn2.
Each of X variables is distributed via the normal distribution with
following parameters:
1. Mean values of all Xi = 0
2. Variances are identical and are equal to
ak2
Find probability density of a random value of Y.

Let Y be the liner combination of the independent random
variables X1 and X2 where Y = X1 -2X2
suppose X1 is normally distributed with mean 1 and standard
devation 2
also suppose the X2 is normally distributed with mean 0 also
standard devation 1
find P(Y>=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?

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