Let X1,X2,.., X25 be i.i.d. random variables from Po(5). Estimate e jjE for the mejdiajan estimator using Monte Carlo estimation.
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- What is the likelihood ratio p(x|C₁) p(x|C₂) in the case of Gaussian densities?Let X be discrete random variable. If P(XLet X be a random variable with density function 1) ={0. k(1 -x), if 0<*<1; otherwise. f(x) Find k, together with the expectation and the variance of the random variable Y defined as Y = 3X - 1.
- Suppose that the length of a phone call in minutes is an exponential random variable with parameter λ=1/10. If someone arrives immediately ahead of you at a public telephone booth, find the probability that you will have to wait, (a) More than 10 minutes (b) Between 10 and 20 minutesSolve in R programming language: 5) Use R to calculate and simulate with the exponential distribution as follows. (a) For an exponential random variable X with λ = 4, simulate 1000 independent exponential random variables by using the R function rexp(n, λ). Calculate the mean and variance of this sample. (b) Compare the empirical results from part (a) with the distribution mean 1/λ and the distribution standard deviation 1/λ.Using the MATLAB Histogram function, "hist.m", Illustrate the Central Limit Theorem by taking two different random variables and show how when appropriately scaled and summed, they converge to a Normal (Gaussian) distribution.
- Consider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(1), x(2), ... , x(n), their true house rents y(1), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x()) = 00 + 01x(e). The parameters 6o and 0, are scalars and are learned by minimizing mean-squared-error loss with L2-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x), y@), n, 0, 01, A. 2. Compute L 3. Compute 4. Write update rules for 6, and O15. The probability distribution of discrete random variable X is given by k for x = 1, 2, 3 x +1 P(X = x) %3DA particular telephone number is used to receive both voice calls and fax messages. Suppose that 20% of the incoming calls involve fax messages, and consider a sample of 20 incoming calls. (Round your answers to three decimal places.) (a) What is the probability that at most 6 of the calls involve a fax message?(b) What is the probability that exactly 6 of the calls involve a fax message?(c) What is the probability that at least 6 of the calls involve a fax message?(d) What is the probability that more than 6 of the calls involve a fax message?
- Write the code that generates a normal sample with given u and o and the code that calculates m and s from the sample. Do the same using the Bayes' estimator assuming a prior distribution for u.PROBLEM 3 The PDF of a random variable X is given in the picture below K is 2 Find the correct value of c. Find the variance of the given random variable. Find P(X<2).Consider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(¹), x(²),...,x(n), their true house rents y(¹), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x)) = 0 + 0₁x). The parameters, and 0₁ are scalars and are learned by minimizing mean-squared-error loss with L1-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x(i),y(i), n, 00, 01, X. 2. Compute L 200 ƏL 3. Compute 20₁ 4. Write update rules for 0o and 0₁ Hint: d|w| dw undefined -1 w>0 w=0 w <0