Question: (Bayesian) Suppose that X is Poisson(λ + 1), and the prior distribution of λ is...

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 distribution we have seen before;

You will have to compute (numerically) the posterior probabilities of the three possible values.

Expert Solution

orchestra answered 1 year ago

Show that the skewness of X~Poisson(λ) is λ^-(1/2)

Determine the Maximum Likelihood Estimator for; 1. λ for the Poisson distribution. 2. θ for the...

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

The number of imperfections in an object has a Poisson distribution with a mean λ =...

The number of imperfections in an object has a Poisson distribution with a mean λ = 8.5. If the number of imperfections is 4 or less, the object is called "top quality." If the number of imperfections is between 5 and 8 inclusive, the object is called "good quality." If the number of imperfections is between 9 and 12 inclusive, the object is called "normal quality." If the number of imperfections is 13 or more, the object is called "bad...

Suppose that X is a Poisson random variable with λ=39. Round your answers to 3 decimal...

Suppose that X is a Poisson random variable with λ=39. Round your answers to 3 decimal places (e.g. 98.765). (a) Compute the exact probability that X is less than 26. (b) Use normal approximation to approximate the probability that X is less than 26. Without continuity correction: With continuity correction: (c) Use normal approximation to approximate the probability that 52<X<78. Without continuity correction: With continuity correction:

For the Poisson Distribution a. Is λ×T a parameter of position, scale, shape or a combination?...

For the Poisson Distribution a. Is λ×T a parameter of position, scale, shape or a combination? Explain. b. We can treat “λ×T” as a single parameter, but they actually represent two parameters: λ and T. Say you stand by the side of the road and count the number of cars that pass by you in one minute. After repeating this process 10 times you find the average number of cars that pass in 10 minutes. In this case, what is...

Suppose that x has a Poisson distribution with μ = 2. (b) Starting with the smallest...

Suppose that x has a Poisson distribution with μ = 2. (b) Starting with the smallest possible value of x, calculate p(x) for each value of x until p(x) becomes smaller than .001. (Round your answers to 4 decimal places.) x 0 1 2 3 4 5 6 7 8 p(x) (d) Find P(x = 2). (Round your answer to 4 decimal places.) (e) Find P(x ≤ 4). (Round your answer to 4 decimal places.) (f) Find P(x < 4)....

Suppose that x has a Poisson distribution with μ = 22. (a) Compute the mean, μx,...

Suppose that x has a Poisson distribution with μ = 22. (a) Compute the mean, μx, variance, σ2xσx2, and standard deviation, σx. (Do not round your intermediate calculation. Round your final answer to 3 decimal places.) µx = , σx2 = , σx = (b) Calculate the intervals [μx ± 2σx] and [μx ± 3σx ]. Find the probability that x will be inside each of these intervals. Hint: When calculating probability, round up the lower interval to next whole...

During lunchtime, customers arrive at Bob's Drugs according to a Poisson distribution with λ = 5...

During lunchtime, customers arrive at Bob's Drugs according to a Poisson distribution with λ = 5 per minute. Show your answers to 3 decimal places. What is the probability of one customer arriving? What is the probability of more than two customers arriving? What is the probability of at most three customers arriving? What is the probability of at least four customers arriving? What is the probability of fewer than two customers arriving?

Let Y denote a random variable that has a Poisson distribution with mean λ = 6....

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).

Both X and S2 are unbiased for the Poisson λ. Which is better? Use the following...

Both X and S2 are unbiased for the Poisson λ. Which is better? Use the following steps to answer this question. a) Generate 200000 random numbers from the Poisson(λ = 2) distribution and arrange them in a matrix with 20 rows. Thus you have 10000 samples of size 20. b) Compute the 10000 sample means and sample variances and store them in objects means and vars, respectively. c) Compute the average of the 10000 sample means and the average of...

Question