Question

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.

Answer 1

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

Explanation:

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`