I have tried to solve number a and b, but I am confused to solve number c. Would you be so kind to help me please? It's about mathematical statistics. I am studying for my midterm. Thank you very much for your help before.

Question

Nora Amin · Accepted Answer

#F(x) = {(0",", x<1),(1 - (9/10)^(n)",", n <= x < n+1", " n in NN):}#

A cumulative distribution function (CDF) #F(x)# gives the probability of a random variable #X# being less than (or equal to) a given value of #x#.

#F(x) = "P"(X<=x)# #color(white)(F(x)) = 1 - "P"(X>x)#

The word "cumulative" indicates

#F(x) = int_(–oo)^xf(t)dt# when #X# is continuous, and

#F(x) = sum_(t=–oo)^x f(t)# when #X# is discrete.

For this #X#, we use

#F(x) = 1 - "P"(X>x)#

#color(white)(F(x)) = 1 - sum_(w=x+1)^oo1/10(9/10)^(w-1)#

#color(white)(F(x)) = 1 - 1/10sum_(w=x)^oo(9/10)^w#

Since all terms in the sum have a common #(9/10)^(x),# we can factor this out, and change the starting point of the sum:

#color(white)(F(x)) = 1 - 1/10(9/10)^(x)sum_(m=0)^oo(9/10)^m#

The infinite sum is now a geometric series, which gives us

#color(white)(F(x)) = 1 - 1/10(9/10)^(x)[1/(1-9/10)]#

#color(white)(F(x)) = 1 - 1/10(9/10)^(x)xx10#

#color(white)(F(x)) = 1 - (9/10)^x#