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# Expected Value (Algebra)

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What happens between 'starting from x=1' to 'using y=x-1'
I know its times by (1-n) and divide by (1-n) - but I don't know why the 'to the power of x' goes to x-1 and how the x within the sum goes to (y+1)..

All help appreciated..
Attached Images

2. (Original post by JKITFC)

What happens between 'starting from x=1' to 'using y=x-1'
I know its times by (1-n) and divide by (1-n) - but I don't know why the 'to the power of x' goes to x-1 and how the x within the sum goes to (y+1)..

All help appreciated..
It's just using the rule of indices:

.

In this case: .

Since is just a constant, you can pull it out of the sum:

3. (Original post by Zacken)
It's just using the rule of indices:

.

In this case: .

Since is just a constant, you can pull it out of the sum:

Perfect - thankyou
4. (Original post by JKITFC)
Perfect - thankyou
And for the next part, think about it just like a "integration substitution"

The sub means that when , we have (that's the bottom limit sorted) and the top limit is so that stays the same.

Then, the means that the and the by re-arranging the sub by adding 1 to both sides of the equation.
5. Thanks - that part clicked just after I posted!

I get the fact that the second part of the addition is a PMF so equals one, and the fact the first part = E(X).
How does (Y+1) go to y(1-n) - is it becasue the (1-n) outside the sum multiplies with the (y+1) inside?
Attached Images

6. (Original post by JKITFC)
Thanks - that part clicked just after I posted!

I get the fact that the second part of the addition is a PMF so equals one, and the fact the first part = E(X).
How does (Y+1) go to y(1-n) - is it becasue the (1-n) outside the sum multiplies with the (y+1) inside?
If you're asking about , then yeah - it's just multipling out, just like ; we have:
7. (Original post by Zacken)
If you're asking about , then yeah - it's just multipling out, just like ; we have:
I mean from the very top line of the second screenshot.
How does (Y+1) go to y(1-n) - so how do they make the (y+1) at the bottom of the first screenshot into the E(X) in the top line of the second screenshot?
8. (Original post by JKITFC)
I mean from the very top line of the second screenshot.
Oh, I think I understand what you mean.

By expanding the where .

So:

9. (Original post by Zacken)
Oh, I think I understand what you mean.

By expanding the where .

So:

Thankyou - exactly what I was looking for - would give rep but wont let me!
10. (Original post by JKITFC)
Thankyou - exactly what I was looking for - would give rep but wont let me!
No problem.

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