R1.
The code to find the average of a random sample of size 40 from the Poisson distribution with = 1.81 is,
mean(rpois(n = 40, lambda = 1.81))
R2.
Use the below command in R to find 10,000 values for
lambdahats = replicate(10000, mean(rpois(n = 40, lambda = 1.81)))
R1. It is easy to show the MME for a Poisson distribution is λ = X....
Find the MME for r and λ for the Gamma distribution given by fX(x; r, λ) = λ r Γ(r) x r−1 e −λx where x > 0, r > 0, and λ > 0. Assume a random sample of size n has been drawn ar-le-k 4. Find the MME for r and λ for the Gamma distribution given by fx(z;r, A) where x > 0, r > 0, and λ 〉 0, Assume a random sample of size n...
R1. Now that you've seen both MME and MLE, we might begin comparing the two worlds. In class, we studied X Uni (0.0) and showed the MLE is aMLE = max Xi One can show the MME for this setup is θΜΜΕ 2 X. As seen in HW2, these estimators are RVs, and each will have its own (sampling) distribution. The sampling distribution gives a good sense of what types of values you'll get from θ when you draw a...
R1. Now that you've seen both MME and MLE, we might begin comparing the two worlds. In class, we studied X Uni (0.0) and showed the MLE is θMLE max Xi. One can show the MME for this set up is all -2 . X. As seen in HW2, these estinators are RVs, and each will have its own (sampling) distribution. The sampling distribution gives a good sense of what types of values you'll get from θ when you draw...
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
R1. Now that you've seen both MME and MLE, we might begin comparing the two worlds. In class, we studied XUnif(0,0) and showed the MLE is OSILE-max Xi. One can show the MME for this setup is OMME -2 X. As seen in HW2, these estimators are RVs, and each will have its own (sampling) distribution. The sampling distribution gives a good sense of what types of values you'll get from θ when you draw a random sample. Use the...
Compute the expected value of the Poisson distribution with parameter λ X ∼ Poisson(λ). Show E[X(X − 1)(X − 2)· · ·(X − k)] = λ ^(k+1) Use this result, and that in question above, to calculate the variance of X
(a) Let YA ~ P(λ) denote a Poisson RV with parameter λ. For a non-random function b(A) > 0, consider the the RVs Xx:-b(A)(YA-A), λ > 0. Use the method of ChFs to find a function b(A) such that XA 1 X as λ 00, where X is a non-degenerate RV. You are expected to establish the fact of convergence and specify the distribution of X ,IE [0,oo)? Explain. (b) Does the distribution of y, converge as ג Hint: (a)...
(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼ Poisson(λ + γ). (b)Let X1, . . . , Xn be an iid random sample from Poisson(λ), provide a sufficient statistic for λ and justify your answer. (c)Under the setting of part (b), show λb = 1 n Pn i=1 Xi is consistent estimator of λ. (d)Use the Central Limit Theorem to find an asymptotic normal distribution for λb defined in part (c), justify...
This is Probability and Statistics in Engineering and Science Please show your work! especially for part B A Poisson distribution with λ=2 X~Pois(2) A binomial distribution with n=10 and π=0.45. X~binom(10,0.45) Question 4. An inequality developed by Russian mathematician Chebyshev gives the minimum percentage of values in ANY sample that can be found within some number (k21) standard deviations from the mean. Let P be the percentage of values within k standard deviations of the mean value. Chebyshev's inequality states...
Suppose X has an exponential distribution with parameter λ = 1 and Y |X = x has a Poisson distribution with parameter x. Generate at least 1000 random samples from the marginal distribution of Y and make a probability histogram.