(1 point) Let ?1 and ?2 have Poisson distributions with the same
average rate λ =...
(1 point) Let ?1 and ?2 have Poisson distributions with the same
average rate λ = 0.6 on independent time intervals of length 1 and
3 respectively. Find Prob(?1+?2=3) to at least 6 decimal
places.
(a) Let Λ = {λ ∈ R : 0 < λ < 1}. For each λ ∈ Λ, let Aλ =
{x ∈ R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ
respectively.
(b) Let Λ = \ {λ ∈ R : λ > 1}. For each λ ∈ Λ, let Aλ = {x ∈
R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ
respectively.
Determine the Maximum Likelihood Estimator for;
1. λ for the Poisson distribution.
2. θ for the Exponential distribution.
Caveat: These are examples of distributions for which the MLE
can be found analytically in terms of the data x1, . . . , xn and
so no advanced computational methods are required and also in each
assume a random sample of size n, x1, x2, . . . , xn
Question:
(Bayesian) Suppose that X is Poisson(λ + 1), and the prior
distribution of λ is binomial(2,1/3).
(a) Find the Bayesian estimate of λ for mean square loss based
on the single observation X, if X = 1.
(b) Find the Bayesian estimate of λ for mean square loss based
on the single observation X, if X = 2
Hints:
Because of its prior distribution, λ can take only three values,
0,1,2.
Don’t expect its posterior distribution to be any...
Let Y denote a random variable that has a Poisson
distribution with mean λ = 6. (Round your answers to three
decimal places.)
(a) Find P(Y = 9).
(b) Find P(Y ≥ 9).
(c) Find P(Y < 9).
(d) Find P(Y ≥
9|Y ≥ 6).
At a train station, international trains arrive at a rate λ = 1.
At the same train station national trains arrive at rate λ = 2. The
two trains are independent.
What is the probability that the first international train
arrives before the third national train?
Starting at time 0, a red bulb flashes according to a Poisson
process with rate λ=1. Similarly, starting at time 0, a blue bulb
flashes according to a Poisson process with rate λ=2, but only
until a nonnegative random time X, at which point the blue bulb
“dies." We assume that the two Poisson processes and the random
variable X are (mutually) independent.
a) Suppose that X is equal to either 1 or 2, with equal
probability. Write down an...
Emails arrive in an inbox according to a Poisson process with
rate λ (so the number of emails in a time interval of length t is
distributed as Pois(λt), and the numbers of emails arriving in
disjoint time intervals are independent). Let X, Y, Z be the
numbers of emails that arrive from 9 am to noon, noon to 6 pm, and
6 pm to midnight (respectively) on a certain day. (a) Find the
joint PMF of X, Y, Z....
2. Let X ~ Pois (λ) λ > 0
a. Show explicitly that this family is “very regular,” that is,
that R0,R1,R2,R3,R4 hold.
R 0 - different parameter values have different functions.
R1 - parameter space does not contain its own endpoints.
R 2. - the set of points x where f (x, λ) is not zero and should
not depend on λ .
R 3. One derivative can be found with respect to λ.
R 4. Two derivatives can...