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

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

So, f(1)=1/c, f(2)=2/c, f(3)=3/c, f(4)=4/c

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

C=10

(b)f(x)=cx, x=1,2,…,10

C+2c+3c+…+10c=1

55c=1

C=1/55

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

2c+4c+6c+8c=1

C=1/20