If we took a random sample of n 10 and observed If we took a random sample of n=10 and observed 2...
Please answer from a-d
Problem 2. Let X be a random variable with one of the following cumulative distribution function. 1.2 1,2 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2,0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 X X Pick the correct cumulative distribution function plot and answer questions: Page 2 of 9 Write down the probability mass function and What is the PMF of X? A. Poisson (3...
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Problem 2. Let X be a random variable with one of the following cumulative distribution function. 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 0.4 F 0.4 0.2 0.2 0.0 0.0 -1.0 -0.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 X Pick the correct cumulative distribution function plot and answer questions: Page 2 of 9 (3 pts) Write down the probability mass function and What is the PMF of...
Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent ran- dom variables, we can find joint probabilities by multiplying together th ndividual probabilities. For example This should remind you the discussion on statistical independence of random variables that can be found in the course book (see page 22) Answer the following questions a...
In class, we showed that the MLE of the mean of a random sample {yi, y2, ..., Yn} from an N(u, 6-) population is exactly the sample average (i.e., û = y). Show that the MLE for the standard deviation is: 1 n @ = – Ëly;-)) ? V ni=1 Hint: Find the partial derivative of the likelihood function with respect to o, then set this equal to zero and solve for o. The result will be a function of...
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 5 sin(x), y 5 cos(x), OS XS/4; about y = -1 V- Sketch the region. 2 1 -1.0 -1.5 -0.5 0.5 1.0 -0.4 -0.2 0.2 0.4 0.6 0.8 1.0 4 3 3 2 2 1 1 -0.5 0.5 110 102 2 04 La Sketch the solid, and a typical disk or washer. y 6 -0.4 -02 0.2...
R codeing simulation
For n = 20, simulate a random
sample of size n from N(µ, 2 2 ), where µ = 1. Note that we just
use µ = 1 to generate the random sample. In the problem below, µ is
an unknown parameter. Plot in different figures: (a) the likelihood
function of µ, (b) the log likelihood function of µ. Mark in both
plots the maximum likelihood estimate of µ from the generated
random sample
(2) For n-20,...
3. Suppose Xi, X2, and X are independent random variables drawn from a binomial distribution with parameters p and n. The observed values are Xi -3, X2-4, and (a) Suppose n 12 and p is unknown. What is the maximum likelihood estimator (b) Suppose p - 0.4 and n is unknown. What is the maximum likelihood estimator for p? for n? (Note: Since n is discrete you can't use calculus for this; just write the formula and use trial and...
Let the random variable X be a random number with the uniform density curve in the figure below. Area = 0.4 Area = 0.5 Area = 0.2 Height = 1 0.3 0.7 0.5 0.8 P(X<0.5 or X > 0.8) P(0.3<X<0.7) (a) (b) Find the following probabilities. P(X 2 0.35) (a) (b) P(X = 0.35) P(0.35 < X < 1.25) (c) P(0.10 < X < 0.20 or 0.6 < X < 0.9) (d) X is not in the interval 0.5 to...
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...
Problem 6: Suppose we observe a random variable X having a binomial distribution with parameters n and zp. (a) What is the generalized likelihood ratio for testing Ho : p-0.5 against H, : p* 0.5? (b) Show that a generalized likelihood ratio test rejects Ho when |X -n/2|2 c. (Hint: it may help to consider the logarithm of the generalized likelihood ratio.) (c) What is the significance level of the test when n 12 and c 5?
Problem 6: Suppose...