Standard Deviation after N trials

[QuadsOverQuads]

Suppose I have a slot machine that pays off according to the following table:

1% : $10
2% : $5
3% : $2
5% : $1
89% : nothing

I understand how to compute the average payout and standard deviation for a single spin.

<u>But:</u> how do I determine the expected standard deviation over an arbitrary number of trials (ie: "N spins")?

Any references/pointers/help would be greatly appreciated.


q/q

[QuadsOverQuads]

Expanding on the question:

Do I need to use a multinomial distribution approach, or can I simply scale the single-spin Std Dev for multiple trials?

Secondly, if I do have to use the multinomial approach, I'm not clear as to how to create a "composite" standard deviation that takes into account all the individual "Xi" standard deviations.

Worst case, I suppose I could create a vector of probabilities and do some sort of repeated self-convolution ... argh, brain starting to hurt again :P

I have done the requisite searches (2p2 and web), but I'm still a few steps away from the answer. If anyone here has a tip or two, or just a good reference to point me to, I'd greatly appreciate it.


q/q

[Beavis68]

the square of the standard dev of one spin is the variance.

the squareroot of the variance for one spin times the numbers of spins in your sample is the SD for that number of spin I think

[QuadsOverQuads]

minor edit:

Assume it's a $1 slot machine.


q/q

[yoshi_yoshi]

well remember that the variance of the sum of two independent variables is just the sum of each variable's variance.

n spins means n independent variables, so you just multiply your original variance by n.

[Beavis68]

isnt that what I said?

[yoshi_yoshi]

beavis, you gotta times the variance by n, not the STDEV.

if you times the STDEV by n, the variance goes up by n^2.

[Beavis68]

that is what I said.

The if you sqare the SD you get variance.

[yoshi_yoshi]

"the squareroot of the variance for one spin times the numbers of spins in your sample is the SD for that number of spin I think"

I interpreted this as:
(STDEV of 1 spin) * n = (STDEV of n spins)

That's wrong and it should be:
(VAR of 1 spin) * n = (VAR of n spins)

[peterchi]

[ QUOTE ]

Do I need to use a multinomial distribution approach, or can I simply scale the single-spin Std Dev for multiple trials?


[/ QUOTE ]
You don't want to use a multinomial distribution here I don't think; that would be if you are trying to model how many of each outcome that you got (i.e. x_1 times you spun $10, x_2 times you spun $5, etc). Rather, we're trying to model your win/loss amount. Someone correct me if I'm wrong.