With given mean and standard deviation we can simulate values for that normal distribution using the excel formula NORMINV(Rand(),mean,sd). This is applicable for X, Y & Z and thus the W value can also be simulated ( W = (X+Y)/Z) The simulation of 50 values are prepared and the histogram with width 3 class interval is shown in the excel sheet. Random Values around the mean and Standard deviation Y Y W Mean 1.. Y.. 0. Deviation Variance Frequency of Simulated Values Bins Frequency

Answer the given question with a proper explanation and step-by-step solution. You will be required to run Expectation Maximization for estimating parameters of a Gaussian mixture model in this example. Refer to lecture slides (Lecture E.2) for the exact formulation. You can use python or do the computations by hand. Recall that you need to use the pdf formulation for computing p(x|theta_k) Consider the following one dimensional data set: 2.3 3.2 3.1 1.6 1.9 11.5 10.2 12.3 8.6 10.9 Assume that we are interested in learning a mixture model with two components (k = 2). Let pi_k denote the probability P(z_i = k) for any i. Let (mu_1,sigma_1) be the parameters for the first Gaussian component of the mixture and (mu_2,sigma_2) be the parameters for the second Gaussian component of the mixture. Given the following initialization: pi_1 = pi_2 = 0.5, mu_1 = mu_2 = 0, and sigma_1 = sigma_2 = 1. Answer the following:       a) After first M step, mu_1 = mu_2 = 6.56     b) After…

Students graduating Atlantis University are being administered a test to check their general competence level at the end of the study program. From a random sample of 100 students, 95 passed and 5 failed. We aim to construct a 95% confidence interval for the proportion p of students at Atlantis University who have achieved a satisfactory competence level after their studies. Answer the following questions: (a) The critical z-value for this problem (the z-value to be used) is z = (give the exact value to TWO decimal places N.xx) (b)The middle of the interval is %. (ROUND to the nearest integer) (c) The error margin is % (use one decimal only and give in terms of percentages).

What is the likelihood ratio p(x|C₁) p(x|C₂) in the case of Gaussian densities?

A 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?

Explain how to use a histogram to estimate the size of a selection of the form σA≤v(r).

For the Covid-19 pandemic, let’s make the following assumptions: (i) the natural infection rate is 20% (i.e., a person who is not vaccinated has 20% chance of being infected); (ii) the vaccine is 90% effective at preventing Covid-19 infection (i.e., a vaccinated person has only 2% chance of being infected); (iii) the probability of a random person being vaccinated is 60%. Without knowing whether or not a person is vaccinated, what is the probability that this person will be infected? Given that a person is infected, what is the conditional probability that this person is vaccinated?

1. The impulse response of a causal system is: h(t) = A cos(wt) e¯¹/¹u(t) where u(t) is the Heaviside step function. The response is measured experimentally with a sampling interval of T. a. Write an expression for the sampled impulse response h[n]. b. Calculate the z transform of h[n] and write an expression for H[z]. Use the tables provided below as necessary. c. Does the system have an infinite impulse response (IIR) or finite impulse response (FIR)? Justify your answer. d. What is the DC gain of H[z]? e. Write a difference equation that describes the output y[n] in terms of input x[n].

"When conducting a binary regression with a skewed predictor, it is often easiest to assess the need for x and log(x) by including them both in the model so that their relative contributions can be assessed directly." Show that indeed the log odds are a function of x and log(x) for the gamma distribution.