Question

Let X be an exponential random variable with parameter λ > 0. Find the probabilities P( X > 2/ λ ) and P(| X − 1 /λ | < 2/ λ) .

Answer #1

Suppose that X|λ is an exponential random variable with
parameter λ and that λ|p is geometric with parameter p. Further
suppose that p is uniform between zero and one. Determine the pdf
for the random variable X and compute E(X).

Let X be a Poisson random variable with parameter λ > 0.
Determine a value of λk that maximizes P(X = k) for k ≥
1.

Let X be a Poisson random variable with parameter λ and Y an
independent Bernoulli random variable with parameter p. Find the
probability mass function of X + Y .

If X is an exponential random variable with parameter λ,
calculate the cumulative distribution function and the probability
density function of exp(X).

Let X be a random variable with an exponential
distribution and suppose P(X > 1.5) = .0123
What is the value of λ?
What are the expected value and variance?
What is P(X < 1)?

A random variable XX with distribution
Exponential(λ)Exponential(λ) has the memory-less
property, i.e.,
P(X>r+t|X>r)=P(X>t) for all r≥0 and
t≥0.P(X>r+t|X>r)=P(X>t) for all r≥0 and t≥0.
A postal clerk spends with his or her customer has an
exponential distribution with a mean of 3 min3 min. Suppose a
customer has spent 2.5 min2.5 min with a postal clerk. What is the
probability that he or she will spend at least an additional 2 min2
min with the postal clerk?

Suppose that X is an exponentially
distributed random variable with λ=0.75. Find each of the
following probabilities
p(x>1)=
p(x>.7)=
p(x<.75)
p(.6<x<3.9)=

Let X1, ..., Xn be a random sample of an
Exponential population with parameter p. That is,
f(x|p) = pe-px , x > 0
Suppose we put a Gamma (c, d) prior on p.
Find the Bayes estimator of p if we use the loss function L(p,
a) = (p - a)2.

Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
λ > 0.
(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
show that E(λ(hat) | ∑ Xi)...

1. Show that if X is a Poisson random variable with parameter
λ, then its variance is λ
2.Show that if X is a Binomial random variable with parameters
n and p, then the its variance is npq.

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