Suppose a discrete random variable X assumes the value 3/2 with probability 0.5 and assumes the value 1/2 with probability 0.5. a). Calculate the expected value and variance of X. b). Suppose that X1, X2, and X3 are three independent random variables with the same distribution as X.What is the expected value of the sum X1 + X2 + X3? The product X1X2X3? Problem 6: Suppose that you have one hundred independent instances of X as defined in Problem 5. Use Markov’s inequality to prove that the probability that the product of all one hundred instances exceeds 50 is less than 1/50.

2. Some properties of Expected value and variance of a random variable. a) Assume that X is an arbitrary discrete random variable, and a and b are constant. Using the definitic Show: and E(aX + b) = a · E(X) + b V(aX + b) = a² · V(X ) Stat 3128 Ali Mahzarnia P STAT 3128 Ali Mahzarnia b) Justify the computational formula of Variance of a random variable which is to justify : V(X) = E[(X – µ°] = Ex – µ)P • ptx) = | 2: Here needs justification By Cauchy Schwarz inequality it can be shown that the right hand side is always positive. Analogs expression in Mechanic : Parallel axis theorem Iem = I– md² Moment of Moment of inetria about Inertia of an an axis shifted object about the center of a mass by d from center of mass (a parallel shift) Icm is dispersion around the mean and is like second central moment (variance) I is like second moment if d is mean m is like sum total all the weight of each of the x which all add up to 1 d squared is like squared of mean since we,

I need help with this:A manufacturer knows that on average 20% of toasters produced require repairs within 1 year after they are sold. When 18 toasters are randomly​ selected, find the smallest number x and largest number y such that​ (a) the probability that at least x of them will require repairs is less than 0.58; ​(b) the probability that at least y of them will not require repairs is greater than 0.84.