For each of the following determine the constant c so that f

For each of the following, determine the constant c so that f(x) satifies the conditions of being a pmf for a random variable X, and then depict each pmf as a line graph:

(a). f(x)=x/c, x=1,2,3,4.

(b). f(x)=cx, x=1,2,3,......,10.

(c). f(x)=c(1/4)^x, x=1,2,3,......

(d). f(x)=c(x+1)², x=0,1,2,3.

(e). f(x)=x/c, x=1,2,3,......,n.

(f). f(x)=c/((x+1)(x+2)), x=0,1,2,3,....

Solution

Note that

Sum(f(x)) = 1

for all pmf.

a)

1/c + 2/c+ 3/c + 4/c = 1

10/c = 1

c = 10 [answer]

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b)

c + 2c + 3c... + 9c + 10c = 1

55c = 1

c =1/55 [answer]

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c)

This is a geometric series with first term c, and common ratio (1/4).

As for a geometric series with first term a1 and common ratio r,

Sum = a1 / (1 - r)

Thus,

Sum = c / (1 - 1/4) = 1

c/(3/4) = 1

c = 3/4 [answer]

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d)

c(0+1)^2 + c(1+1)^2 + c(2+1)^2 +c(3+1)^2 = 1

29c = 1

c = 1/29 [answer]

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