Question

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?

Answer #2

answered by: anonymous

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}...

1)Let X1, ..., Xn be independent standard normal random
variables, we know that X2 1 + ... + X2 n follows the chi-squared
distribution of n degrees of freedom. Find the third moment of the
the chi-squared distribution of 2 degrees of freedom.
2) Suppose that, on average, 1 person in 1000 makes a numerical
error in preparing his or her income tax return. If 10,000 returns
are selected at random and examined, find the probability that 6 or
7...

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 , X3 ,
X4 be a random sample of size 4 from a geometric
distribution with p = 1/3.
A) Find the mgf of Y = X1 + X2 +
X3 + X4.
B) How is Y distributed?

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,... 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

Suppose x1, x2, x3, x4 is linearly independent in V . Prove that
x1 − x2, x2 − x3, x3 − x4, x4 is also linearly independent in V

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?

Let X1, X2, X3, and X4 be a random sample of observations from
a population with mean μ and variance σ2.
Consider the following estimator of μ:
1 = 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4. Is this a biased
estimator for the mean? What is the variance of the estimator? Can
you find a more efficient estimator?) ( 10 Marks)

Let X1, X2 , X3 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 X1, X2, X3 and compute P(Y >
1000).

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