Question

f(x)=cx for x=1,2,...,n find c so that f(x) is a valid probability function , then derive the mean of X

Answer 1

For this to be a valid PDF, you must satisfy
\[\large\sum_{x=1}^n f(x)=\sum_{x=1}^ncx=1\]

Answer 2

Recall the formula,
\[\large\sum_{k=1}^nk=\frac{n(n+1)}{2}\]
Using this gives
\[\large\sum_{x=1}^ncx=c\sum_{x=1}^nx=\frac{cn(n+1)}{2}=1~~\iff~~c=\frac{2}{n(n+1)}\]

Answer 3

Cheers for that SithsAndGiggles, have been stuck on that a while- need to do some catchup work ! appreciate it

Answer 4

You're welcome!

The part with the mean is fairly simple. The mean is given by the expected value formula,
\[E(X)={\large\sum_{x=1}^nxf(x)}\]
You might find this formula useful:
\[\large\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}\]

Answer 5

sorry for my ignorance, but could you possibly show me how you would derive the mean? (will let me know if i did it correctly) thanks

Answer 6

As per the formula for expected value,
\[\large\begin{align*}E(X)&=\sum_{x=1}^nxf(x)\\
&=\sum_{x=1}^nx(cx)\\
&=c\sum_{x=1}^nx^2\\
&=c\frac{n(n+1)(2n+1)}{6}\\
&=\frac{2}{n(n+1)}\cdot\frac{n(n+1)(2n+1)}{6}\\
&=\frac{2n+1}{3}\\

\end{align*}\]