NOTE: DO Part b) ONLY 2. A discrete random variable X has the following pmf: A...
NOTE: DO PART b) ONLY 2. A discrete random variable X has the following pmf: A random sample of size n = 30 produced the following observations 1,3,0,00, 2, 22,0,1,2,0 1,1,0,1,1, 3, î021, 3. i 20.3.0, 2, i, (a) (i) Find and s for this sample. (ii) Find E(X) and var(X) (iii) Find the method of moments estimate of θ iv) Find the standard error of this estimate. (b) (i) Find the likelihood function (ii) Show that the inaximum likelihood...
2. A discrete random variable X has the following pmf A random sample of size n 30 produced the following observations: (a)) Find and s for this sample Find E(X) and var(X) (iii) Find the method of moments estimate of θ (iv) Find the standard error of this estimate. (b) (i) Find the likelihood function (ii) Show that the maximum likelihood estimate of θ is -1 fo/n, where fo is the number of observed 0's in the sample. (iii) Find...
2. (Discrete uniform). Consider the PMF P(X x)= for x 1,2,...0 _ You have a random sample of size three from this distribution: {2,3,10}. a. Find the method of moments estimate for 0 HINT: a very useful fact is that k1 n(n+1) 2 b. Find the MLE for 0 c. Which estimator is unbiased? d. Which estimator is preferred? 2. (Discrete uniform). Consider the PMF P(X x)= for x 1,2,...0 _ You have a random sample of size three from...
3. Let X be a discrete random variable with the following PMF: 0.1 for x 0.2 for 0.2 for x=3 Pg(x)=〈 0.1 for x=4 0.25 for x=5 0.15 for x=6 otherwise a) (10 points) Find E[X] b) (10 points) Find Var(X) c) Let Y-* I. (15 points) Find E[Y] II. (15 points) Find Var(Y) X-HX 4. Consider a discrete random variable X with E [X]-4x and Var(X) = σ. Let Y a. (10 points) Find E[Y] b. (20 points) Find...
2. A discrete random variable X has the following pmf: p(x)| 1-8 30/4 θ/4 A random sample of size n 30 produced the following observations:
Can you explain how to do parts a-c? 4. Suppose that X is a discrete random variable with 2 P(X 0) Chapter 8 Estimation of Parameters and Fitting of Probability Distributions P(X = 1) = ) 2 P(X = 3) =-(1-9) where 0 θ 1 is a parameter. The following 10 independent observati were taken from such a distribution: (3, 0, 2, 1, 3, 2, 1, 0, 2, 1). a. Find the method of moments estimate of e. b. Find...
Let Xi,... , Xn be a random sample from a normal random variable X with E(X) 0 and var(X)-0, i.e., X ~N(0,0) (a) What is the pdf of X? (b) Find the likelihood function, L(0), and the log-likelihood function, e(0) (c) Find the maximun likelihood estimator of θ, θ (d) Is θ unbiased?
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
Assume that you have random variable X with pdf or pmf f(x; θ1, . . . , θk). Let X1, . . . , Xn be a random sample from X. Then Mj = (1/n)Xn i=1 (Xi)j is known as the j-th sample moment of the sample. The moment estimators of θ1, . . . , θk, denoted by ˜θ1, . . . , ˜θk, are the values of θ1, . . . , θk which solve the k equations...
(9) [12 pts] An exponentially distributed random variable, call it X, has the following probability density functior f(x)-oe ex , x > 0, θ > 0 Note that ElX] and VX]ー1 For the rest of this question, assume that you have a data set (xn1 consisting of a random sample of N observations of X. (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that...