Question

et Y = X1 + X2 + 2X3 represent the perimeter of an isosceles trapezoid, where X1 is normally distributed with a mean of 113 cm and a standard deviation of 5 cm, X2 is normally distributed with a mean of 245 cm and a standard deviation of 10 cm, and X3 is normally distributed with a mean of 98 cm and a standard deviation of 2 cm.

1. Find the mean perimeter of the trapezoid.

2. Suppose X1, X2, and X3 are independent of each other. Find the variance of the perimeter of the trapezoid.

3. Suppose that the covariance between X1 and X2 is 35, but X3 is independent of both X1 and X2. Find the variance of the perimeter of the trapezoid.

Answer #1

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) ?

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}...

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]

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, and X3 represent the times necessary to perform three
successive repair tasks at a certain service facility. Suppose they
are independent, normal rv's with the same expected value μ=56 and
variance σ^2=20. Let T0=X1+X2+X3.
(a) Calculate P(T0≤183)P(T0≤183)
(b) Calculate P(153≤T0≤183)P(153≤T0≤183)
(c) Calculate P(x¯>51)P(x¯>51)
(d) Calculate P(54≤x¯≤58)P(54≤x¯≤58)

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?

Suppose that X1 and X2 are independent standard normal random
variables. Show that Z = X1 + X2 is a normal random variable with
mean 0 and variance 2.

The permeability of a membrane used as a moisture barrier in a
biological application depends on the thickness of three integrated
layers. Let X1 denote the thickness of layer 1,
X2 denote the thickness of layer 2, and X3
denote the thickness of layer 3. X1, X2, and
X3 are normally distributed with means of 1.1, 1.3, and
1.5 millimeters, respectively. The standard deviations are 0.1,
0.3, and 0.5 millimeters, respectively.
a Suppose X1, X2, and X3 are
independent. Determine...

Suppose X1 is from a population with mean µ and variance 1 and
X2 is from a population with mean µ and variance 4 (X1, X2 are
independent). Construct an estimator of ? as ?̂=??1+(1−?)?2. Show
that ?̂ is unbiased for ?. Find the most efficient estimator in
this class, that is, find the value of a such that the estimator
has the smallest variance.

Suppose X1 is from a population with mean µ and variance 1 and
X2 is from a population with mean µ and variance 4 (X1, X2 are
independent). Construct an estimator of ? as ?̂=??1+(1−?)?2. Show
that ?̂ is unbiased for ?. Find the most efficient estimator in
this class, that is, find the value of a such that the estimator
has the smallest variance.

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