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- Q1 (a) A fair coin is tossed 5 times and the random variable X counts the number of headsthat come in the 5 tosses. Use the definition of expected value for discrete randomvariables to compute the expected value of X. (b) The continuous random variable T has f(t) = 2/t3t ≥ 10 t < 1as its probability densityfunction. Use the definition of expected value for continuous random variables tocompute the expected value of T.Exercise 6 - Beta Distributions Suppose X~Beta(a, B) with X c (0, b]: 1 P(x; a, B) : xa-'(b – B(a, 8)ba+B-1 **-(b - x)8-1 a) Describe the shape of the probability distribution for the following shape parameters 1. a = B = 1 2. a = B = 0.25 3. a = B = 3 b) Under which conditions for the shape parameters would the beta distribution be asymmetric c) What would distinguish the Kullback-Leibler divergence from the absolute difference between X-Beta(1,1) and X~Normal(0.5 + b,0)?Could you solve (g),(h)? Thank you.
- PDF = f(x) 2/x^3 for x>1 Question: - what is Probability ... P(x<4 or x>8)? - x such that P(X>x) ?Compute P(X) using the binomial probability formulaThen determine whether the normal distribution can be ed to estimate this probabilityIf soapproximate P(X) using the normal distribution and compare the result th the exact probability n=53, p = 0.7 and X = 41Plz don't give solution in image format.
- A mass function (PMF) for a discrete variable X € {1,2,3} is given by the table X12 3 Let u designate the expected value u = E(X) and o? = Var(X) the variance The moment generating function is given by p(s) = E(e³*) = > P;e*i We suppose that p(s) have the maclaurin series P(s) = ao + a,s + azs2 + azs3+... Further we suppose that log o(s) have the maclaurin series logø (t) = bo + bs + b2s2+... Which of the following statements are true 1. µ = 1.75 2. p(t) = 1/2 elt + 1/4 · e2t + 1/4 e3t 3. bo = 1 4. az = 0.29PLEASE SHOW COMPLETE SOLUTIONContinuous random variables 5. A charity group raises funds by collecting waste paper. A skip-full will contain an amount, X of other materials such as plastic bags and rubber bands. X may be regarded as a random varibales with probability density funnction. All nmerical values are in units of 100kg. Calculate c. express your answer in fraction c(x - 1)(4x) f(x) = _{[C(x − 1) for 1SEE MORE QUESTIONS
