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- Assume that you have random variable X with the following probability density function with a > 0, b > 0: Г(а+) I(a)T(b) (1 – x)b-1 if € (0, 1) -1 fx(x) = otherwise. (a) Suppose that a = 2 and b = 2. In this case, derive E[X] and var[X]. (b) Continuing with the X from part (a) with a = 2 and b = 2, let Z|X = x ~ Binomial(n, x). Find E[Z] and var[Z]. (c) Find the probability density function of Y defined by the following transfor- mation: Y = 1— ХThe United States Department of Agriculture (USDA) found that the proportion of young adults ages 20–39 who regularly skip eating breakfast is 0.2380.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size ?=500n=500 of young adults ages 20–39 in the United States. Apply the central limit theorem to find the probability that the number of individuals, ?,X, in Lance's sample who regularly skip breakfast is greater than 126126. You may find table of critical values helpful. Express the result as a decimal precise to three places. Then, Apply the central limit theorem for the binomial distribution to find the probability that the number of individuals in Lance's sample who regularly skip breakfast is less than 9898. Express the result as a decimal precise to three places.7. If a random variable has the probability density function -1≤x≤3 otherwise f(x)=k(x²-1) =0If P(B) = ² and P(A'^B) = 12, for two events A and B, find P(A/B).The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…X is a normally normally distributed variable with mean u =10 and standard deviation a =4. Find A) P(x 1) C) P(10Find the expected value of the continuous random variable g(x)= x^2+3x+2, if the density function for random variable x are shown in the following equation. f() = [F(2x-1) 0Question 1.2 Consider the function f (x) = (1/24(x^2 +1) 1 < or = x < or = 4) = (0 otherwise) Calculate P (x = 3) Calculate P (2 < or = x < or = 3) Question 1.3 Consider the function f (x) = (k - x/4 1 < or = x < or = 3) = (0 otherwise) which is being used as a probability density function for a continuous random variable x? a. Find the value of K b. Find P (x < or = 2.5)Determine the probability density function of the random variable X* if x 1 ON|30Let X be a continuous random variable with range [−ln5,0] and its probability density function is given by the following function: f(x)=ce^−x, where cc is a constant. (1) Find the value of c . Answer: (2) Find the probability P(−ln2≤X≤0) . Answer:The probability density function of the random variable X is as in the picture with λ> 0. Investigate whether the most likelihood estimator (λ^) of λ is neutral. hint : ?(?) = √??/2 and ?(?²)=2?SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON