Question

F(x) = (128/127)(1/2)x, x = 1,2,3,...7. determine the requested values: round your answers to three decimal places (e.g. 98.765). (a)p(x ≤ 1) (b)p(x > 1) (c) mean (d) variance

Answer 1

a.

`\mathbb P(X\le 1)=\mathbb P(X=1)=(128)/(127)\left(\frac12\right)^1=(64)/(127)`

b.

`\mathbb P(X>1)=1-\mathbb P(X\le1)=1-(64)/(127)=(63)/(127)`

c.

`\mathbb E(X)=\displaystyle\sum_(x=1)^7 x\,f_X(x)=(64)/(127)\sum_(x=1)^7 x\left(\frac12\right)^(x-1)`

Suppose
`f(y)=\displaystyle\sum_(x=0)^7 y^x`. Then
`f'(y)=\displaystyle\sum_(x=1)^7 xy^(x-1)`. So if we can find a closed form for
`f(y)`, in terms of
`y`, we can find
`\mathbb E(X)` by evaluating the derivative of
`f(y)` at
`y=\frac12`.


`f(y)=\displaystyle\sum_(x=0)^7 y^x=y^0+y^1+y^2+\cdots+y^6+y^7`

`y\,f(y)=y^1+y^2+y^3+\cdots+y^7+y^8`

`f(y)-y\,f(y)=y^0-y^8`

`(1-y)f(y)=1-y^8`

`f(y)=(1-y^8)/(1-y)`

`\implies f'(y)=(7y^8-8y^7+1)/((1-y)^2)`

`\implies\mathbb E(X)=(64)/(127)f'\left(\frac12\right)=(64)/(127)*(247)/(64)=(247)/(127)`

d.

`\mathbb V(X)=\mathbb E(X^2)-\mathbb E(X)^2`

We find
`\mathbb E(X^2)` in a similar manner as in (c).


`\mathbb E(X^2)=\displaystyle\sum_(x=1)^7 x^2\,f_X(x)=(32)/(127)\sum_(x=1)^7x^2\left(\frac12\right)^(x-2)`

Now,


`f(y)=\displaystyle\sum_(x=0)^7y^x`

`\implies f'(y)=\displaystyle\sum_(x=1)^7xy^(x-1)`

`\implies f''(y)=\displaystyle\sum_(x=2)^7x(x-1)y^(x-2)`

We know that


`f''(y)=-(42y^8-96y^7+56y^6-2)/((1-y)^3)`

`\implies f''\left(\frac12\right)=(219)/(16)`

We also have


`f''(y)=\displaystyle\sum_(x=2)^7x(x-1)y^(x-2)`

`f''(y)=\displaystyle\sum_(x=2)^7x^2y^(x-2)-\sum_(x=2)^7xy^(x-2)`

`f''(y)=\displaystyle\frac1{y^2}\left(\sum_(x=2)^7x^2y^x-\sum_(x=2)^7xy^x\right)`

`f''(y)=\displaystyle\frac1{y^2}\left(\bigg(\sum_(x=1)^7x^2y^x-y\bigg)-\bigg(\sum_(x=1)^7xy^x-y\bigg)\right)`

`f''(y)=\displaystyle\frac1{y^2}\left(\sum_(x=1)^7x^2y^x-\sum_(x=1)^7xy^x\right)`

so that when
`y=\frac12`, we get


`(219)/(16)=4\left((127)/(128)\mathbb E(X^2)-(127)/(128)\mathbb E(X)\right)\implies\mathbb E(X^2)=(685)/(127)`

Then


`\mathbb V(X)=(685)/(127)-\left((247)/(127)\right)^2=(25,986)/(16,129)`