Assume that there is a computer program that produces random numbers from Unif (0,1). Explain how random numbers could be produced from the pdf f) = {4y*e-y*, 0< y 0, otherwise (Hint: Remember probability integral transformation)
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- Suppose that the probability of making a mistake in translation at each translational step is a small number, δ. Show that the probability, p, that a given protein molecule, containing n residues, will be completely error-free is (1 – δ)n.Suppose that random variables X and Y are defined on a sample space with only two elements. Suppose that Cov(X, Y ) = 0. Prove that X and Y are independent.a cereal company claims that the mean weight of the cereal in its packets is at least 14oz. express the null hypothesis and the alternatvie hypothesis in symbolic form
- Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variablesLet Mx, y be the moment generating function of random variables that are not independent of X and Y. Which of the following / which are not the properties of the function Mx, y?Here we have a random walk (with p=1/2) on the integers in the range [0,7] (endpoints 0 and 7 are absorbing). With a full explanation, suggest the expected number of moves to absorption, starting at x0=4.
- Construct a random variable X such that its first 2n moments exit and rest of moments don’t exit.Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?Let X1, X2, ..., Xn be independent and identically distributed random variables, X ∼ Exp(λ). Show that the sum X1 + X2 + · · · + Xn is Γ(n, λ) distributed.