O Cross entropy loss function for a logistic regression based model is given as: Cost = (Vactual) In (Ypredcted) - (1 - Yactual) ln(1 - Ypredcted) Consider a case having two instances shown below having Yactual v/s Ypredicted Yactual 1 1 Ypredicted 0.95 0.1 For these instances, verify the fact that using the cross entropy loss function penalizes the "misclassified case" more in comparison to the mean square error loss function.

Assume the following simple regression model, Y = β0 + β1X + ϵ ϵ ∼ N(0, σ^2 )   Now run the following R-code to generate values of σ^2 = sig2, β1 = beta1 and β0 = beta0. Simulate the parameters using the following codes:   Code:   # Simulation ##  set.seed("12345")  beta0 <- rnorm(1, mean = 0, sd = 1) ## The true beta0  beta1 <- runif(n = 1, min = 1, max = 3) ## The true beta1  sig2 <- rchisq(n = 1, df = 25) ## The true value of the error variance sigmaˆ2    ## Multiple simulation will require loops ##  nsample <- 10 ## Sample size n.sim <- 100 ## The number of simulations  sigX <- 0.2 ## The variances of X #   # Simulate the predictor variable ##  X <- rnorm(nsample, mean = 0, sd = sqrt(sigX))   Q1  Fix the sample size nsample = 10 . Here, the values of X are fixed. You just need to generate ϵ and Y . Execute 100 simulations (i.e., n.sim = 100). For each simulation, estimate the regression coefficients (β0, β1) and the error variance (σ 2 ). Calculate the mean of…

Given a two-category classification problem under the univariate case, where there are two training sets (one for each category) as follows: D₁ = (-3,-1,0,4} D₂ = {-2,1,2,3,6,8} Given the test example x = 5, please answer the following questions: have and a) Assume that the likelihood function of each category has certain paramétric form. Specifically, we p(x | w₁) N, 07) p(x₂)~ N(μ₂, 02). Which category should we decide on when maximum-likelihood estimation is employed to make the prediction?

In R, you will use simulations to prove that the binomial distribution is correct. Recall that the binomial distribution has two parameters n and p. There are n trials and each has two possible outcomes, with probability p for “success” and 1-p for “failure”. The binomial gives the probability distribution for the number of successes in n trials. You will conduct simulations with r replicates, where each simulation replicates does n simulated “coin flips”. You will add up the number of successes in each coin flip, and compare the result to the true distribution: Generate n*r values from the uniform(0,1) distribution and arrange these in an rxn matrix. Each value less than p is considered a “success”. For each row from part I, count the number of successes. The number of possible successes ranges from 0 to n. Use the table function in R and the value_counts function in Python and to count up the number of replicates with each number of successes. Make a table that compares the…