In an Accept–Reject algorithm that generates a N (0, 1) random vari- able from a double-exponential distribution with density g(x|α) = (α/2) exp(−α|x|), compute the upper bound M over f /g and show that the choice α = 1 optimizes the corresponding acceptance rate.
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In an Accept–Reject
able from a double-exponential distribution with density g(x|α) = (α/2) exp(−α|x|),
compute the upper bound M over f /g and show that the choice α = 1 optimizes the
corresponding acceptance rate.
Step by step
Solved in 2 steps
- What is the likelihood ratio p(x|C₁) p(x|C₂) in the case of Gaussian densities?It is known that a natural law obeys the quadratic relationship y = ax“. What is the best line of the form y = px + q that can be used to model data and minimize Mean-Squared-Error if all of the data points are drawn uniformly at random from the domain [0,1]? r* ur, a,Computer Science Markov's inequality for probabilities. Let p(x) be a probability mass function. Prove, for all d > 0, that Pr {p(X) < d} log 1 d < H(X). ECE or CS
- A random variable X with two-sided exponential distribution given by has moment generating function given by M X (t)= e^ t +e^ -t -2 t^ 2 . f x (x)= x+1,&-1\\ 1-x,&0<= x<=1 - 1 <= x <= 0 (a) Using M_{X}(t) or otherwise, find the mean and variance of X. (b) Use Chebychev inequality to estimate the tail probability, P(X > delta) , for delta > 0 and compare your result with the exact tail probability.4. Given a random variable w with den sity F(w) and a random variable p as the umiform im the interval (-71, +71) where wlP (two w and p are independent) R.V.S Suppose the stochastic process, xt) =a Cos (wt+P). meam a) show that xlt) is wss with zero and auto correlation equal R(E) = E (Coswr) j(wt+P) b) show that zt)=aé is also a Wss.Let 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.
- 2. Use the rbinom() function to generate a random sample of size N = 50 from the bino- mial distribution Binomial(n, p), with n 6 and p = 0.3. Note that this distribution has mean u = np and standard deviation o = Vnp(1 – p). Record the obtained sample as a vector v. Repeat the tasks of Problem 1 for the sample v.For the transition probability matrix P = 1 2 3 4 5 1 0.2 0.1 0.15 0 0.55 2 0 1 0 0 0 3 0.35 0.2 0.2 0.1 0.15 4 0 0 0 1 0 5 0.25 0.2 0.15 0.25 0.15 (a) Rewrite P in the canonical form, clearly identifying R and Q. (b) For each state, i, calculate the mean number of times that the process is in a transient state j, given it started in i. (c) For each state i, find the mean number of transitions before the process hits an absorbing state, given that the process starts in a transient state i. (d) For each state i, find the probability of ending in each of the absorbing states.Consider a test of H0 : μ ≤ 100 versus H1 : μ > 100. Suppose that a sample of size 20 has a sample mean of X = 105. Determine the p-value of this outcome if the population standard deviation is known to equal (a) 5 (b) 10
- Prove that I(X; Y |Z) ≥ I(X; Y ) . Note: X, Y, and Z are random variables. X and Z are independent.What is the inverse CDF of the uniform distribution from 0 to 1? Write your answer in terms of u. E.g. the inverse of the CDF of the exponential distribution is -1/lambda*log(1-u)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 O1