Question

Given an exponential random variable X with parameter θ and θ is uniformly distributed on the interval (2,4). Solve for the expected value and variance of X.

Answer #1

. Properties of the uniform, normal, and exponential
distributions Suppose that x₁ is a uniformly distributed random
variable, x₂ is a normally distributed random variable, and x₃ is
an exponentially distributed random variable. For each of the
following statements, indicate whether it applies to x₁, x₂, and/or
x₃. Check all that apply. x₁ x₂ x₃ (uniform) (normal) (exponential)
The probability that x equals its expected value is 0. The
probability distribution of x has two parameters. The mean and
standard...

The random variable X is uniformly distributed in the interval
[0, α] for some α > 0. Parameter α is fixed but unknown. In
order to estimate α, a random sample X1, X2, . . . , Xn of
independent and identically distributed random variables with the
same distribution as X is collected, and the maximum value Y =
max{X1, X2, ..., Xn} is considered as an estimator of α.
(a) Derive the cumulative distribution function of Y .
(b)...

The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0. (Please note the equation includes the term
-(x/θ)2 )
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Cramer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).

Included all steps. Thanks
The random variable X is uniformly distributed in the interval
[0, α] for some α > 0.
Parameter α is fixed but unknown. In order to estimate α, a
random sample X1, X2, . . . , Xn of independent and identically
distributed random variables with the same distribution as X is
collected, and the maximum value Y = max{X1, X2, ..., Xn} is
considered as an estimator of α.
(a) Derive the cumulative distribution function...

Suppose that X is a random variable uniformly distributed over
the interval (0, 2), and Y is a random variable uniformly
distributed over the interval (0, 3). Find the probability density
function for X + Y .

The working life (in years) of a certain type of machines is an
exponential random variable with parameter λ, which depends on the
quality of its chip. Suppose that the quality of chips are random
in a sense that λ is uniformly distributed between [0.5, 1].
1.Find the expected working life of a machine.
2.Find the variance of the working time of a machine.

Suppose that X is uniformly distributed on the interval [0,5], Y
is uniformly distributed on the interval [0,5], and Z is uniformly
distributed on the interval [0,5] and that they are
independent.
a)find the expected value of the max(X,Y,Z)
b)what is the expected value of the max of n independent random
variables that are uniformly distributed on [0,5]?
c)find pr[min(X,Y,Z)<3]

Let X ∼ Geo(?) with Θ = [0,1].
a) Show that pdf of the random variable X is in the
one-parameter
regular exponential family of distributions.
b) If X1, ... , Xn is a sample of iid Geo(?) random variables
with
Θ = (0, 1), determine a complete minimal sufficient statistic
for ?.

Let we have a sample of 100 numbers from exponential
distribution with parameter θ
f(x, θ) = θ e- θx , 0
< x.
Find MLE of parameter θ. Is it unbiased estimator? Find unbiased
estimator of parameter θ.

X and Y are independent. X is Rayleigh random variable with a
parameter 5 and y is exponential with a parameter of 5. Obtain the
mean of and variance of Z=4X+6Y

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