4. If X is an integer-valued random variable with characteristic function ø, show that 1 P(X = k) = - 27 -itk o (1) dt. What is the corresponding result for a random variable whose distribution is arithmetic with span 2 (that is, there is probability one that X is a multiple of 2, and 1 is the largest positive number with this property)?

Q: 2] If a random variable X is defined such that E[(X - 1)2]=10 ,E[(X- 2)²]=6. find u and o?. | =

A:

Q: 4. (a) Consider a nonnegative, integer valued random variable Y, and a random sample X1, ·.. , Xn.…

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A: All your answers are correct except (b) and (c)  P(X≤3) So, I provide the answers to those only.

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Q: Suppose x has a distribution with u = 18 and o = 14. (a) If a random sample of size n = 46 is drawn,…

A: According to the provided information, Mean (μ) = 18 Standard Deviation (σ) = 14

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Q: 24. Let X1, X2, .., X 100 be a random sample of size 100 from a distribution with density S(2) = {6…

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Q: 1. For a nonnegative integer-valued random variable Y, show that EiP(Y >i) =LE(Y*)– E(Y)].

A: The expectation formula is as follows:

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Q: Example 3: If x1, X2 , ... , Xn is a random sample from a normal population .2 N(µ,1), show that t=-…

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Q: (4) Let X be a discrete random variable with domain D. Prove that Var(X) = E(X?) – E(X).

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Q: 4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o². Show that E(X(X-1)]=µ(µ-1)+o?

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A: The PMF of the variable X is given below: The PMF of the discrete random variable X is: