Please 

mutheu.stella01

mutheu.stella01

Answered question

2022-05-02

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Answer & Explanation

Don Sumner

Don Sumner

Skilled2023-05-04Added 184 answers

The set given is {(1/x,x=1,2,3,...k),(0,otherwise)}, where k is a positive integer. To find the MGF of Y=2X, we need to first find the PMF of Y.
Since Y=2X, the possible values of Y are even integers between 0 and 2k, inclusive. Let y be such an even integer. Then, we have:
P(Y=y)=P(2X=y)=P(X=y/2)
If y=0, then P(Y=0)=P(X=0)=0 since X takes only positive values. If y=2,4,,2k, then we have:
P(Y=y)=P(X=y/2)={1/kif y=2,4,,k0otherwise
Thus, the PMF of Y is given by:
PY(y)={1/kif y=2,4,,k0otherwise
Now, let's find the MGF of Y. We have:
MY(t)=E(etY)=y=02ketyPY(y)
=e0PY(0)+y=2kety1k
=1ky=0ke2ty
=1k(1+e2+e4++e2kt)
This is a finite geometric series with first term 1 and common ratio e2, so we can use the formula for the sum of a finite geometric series to get:
MY(t)=1k·1e2(k+1)t1e2t
Therefore, the MGF of Y is:
MY(t)=1e2(k+1)tk(1e2t)

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