Question

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}.

Answer #1

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

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 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,... 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 X1 + X2 +···+
XN2?
(A) N2
(B) λ + λ2
(C) λ2
(D) 1/λ2

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)

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, . . . , Xn are independent identically
distributed random
variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and
Y3 = X1 + X2. Find the following : (in terms of σ2)
(a) Var(Y1)
(b) cov(Y1 , Y2 )
(c) cov(X1 , Y1 )
(d) Var[(Y1 + Y2 + Y3)/2]

Consider n independent variables, {X1, X2, . . . , Xn} uniformly
distributed over the unit interval, (0, 1). Introduce two new
random variables, M = max (X1, X2, . . . , Xn) and N = min (X1, X2,
. . . , Xn).
(A) Find the joint distribution of a pair (M, N).
(B) Derive the CDF and density for M.
(C) Derive the CDF and density for N.
(D) Find moments of first and second order for...

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 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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