The expected value of a Geometric Series

Click For Summary
The discussion focuses on proving the expected value of a geometric distribution, expressed as μ = 1/p, without using moment generating functions. The user begins with the definition of expected value and manipulates the summation but encounters difficulty in evaluating it. Participants suggest differentiating the summation formula for y^r to derive a solution. Clarifications are made regarding the proper handling of factors in the summation process. The conversation emphasizes the importance of correctly applying differentiation to evaluate the series involved in the expected value calculation.
relinquished™
Messages
79
Reaction score
0
I'm supposed to prove that in a geometric distribution, the expected value,

<br /> \mu = \frac{1}{p}<br />

without the use of moment generating functions (whatever that is)

I start off with the very definition of the expected value.

<br /> \mu_x = E(x) = \sum x \cdot p \cdot (1-p)^{x-1}<br />

<br /> \mu_x = p \sum x \cdot (1-p)^{x-1}<br />

<br /> \mu_x = p \sum x \cdot (1-p)^x \cdot (1-p)^{-1}<br />

<br /> \mu_x = \frac{p}{1-p} \sum x \cdot (1-p)^x<br />

Now I get stuck because I don't know how to evaluate the summation. Can anyone help me out?

btw, x starts from 1 to n
 
Physics news on Phys.org
can you sum y^r as r goes from 1 to n. what if you differentiate both sides?
 
I am assuming that y^r is (1-p)^x. If I would convert the summation into its series and differentiate both sides, what would be the derivative of \mu_x?
 
erm, what? i indicated to you how to sum a certain kind of series, the series you wanted to sum. I'm not doing anything with differentiating mu_x.
 
err... sorry, my bad. So, when you said differentiate both sides I thought both sides of the equation. What you really mean is that in order to evaluate the summation you need to differentiate, am I understanding it right?
 
you know a formula :

S(n) = sum 1 to n of y^r

that is anequation in y, diff wrt to y and you'll find a formula for a sum that looks a lot like the one you want to sum in your problem. you've pulled that factor of 1/(1-p) out when you shouldn't have: it'll make it more transparent when you put it back in.
 

Thread 'Please Explain (actually explain) The Monty Hall Problem'

Hello, I'm joining this forum to ask two questions which have nagged me for some time. They both are presumed obvious, yet don't make sense to me. Nobody will explain their positions, which is...uh...aka science. I also have a thread for the other question. But this one involves probability, known as the Monty Hall Problem. Please see any number of YouTube videos on this for an explanation, I'll leave it to them to explain it. I question the predicate of all those who answer this...
View full post »

Thread 'A variant of the Monty Hall problem'

There is a nice little variation of the problem. The host says, after you have chosen the door, that you can change your guess, but to sweeten the deal, he says you can choose the two other doors, if you wish. This proposition is a no brainer, however before you are quick enough to accept it, the host opens one of the two doors and it is empty. In this version you really want to change your pick, but at the same time ask yourself is the host impartial and does that change anything. The host...
View full post »

Thread 'Value of intuitionistic logic'

I'm taking a look at intuitionistic propositional logic (IPL). Basically it exclude Double Negation Elimination (DNE) from the set of axiom schemas replacing it with Ex falso quodlibet: ⊥ → p for any proposition p (including both atomic and composite propositions). In IPL, for instance, the Law of Excluded Middle (LEM) p ∨ ¬p is no longer a theorem. My question: aside from the logic formal perspective, is IPL supposed to model/address some specific "kind of world" ? Thanks.
View full post »

Similar threads

I Expected value of X~Geom(p) given X+Y=z
Replies
5
Views
2K
MHB Distribution, expected value, variance, covariance and correlation
Replies
14
Views
2K
MHB Probability to get a chocolate snowman or a chocolate reindeer
Replies
2
Views
2K
MHB For which values are the variables independent?
Replies
4
Views
1K
MHB How Many Cafes Should Be Surveyed for Accurate Daily Turnover Estimates?
Replies
10
Views
3K
I How to obtain moment bound from the importance sampling identity?
Replies
1
Views
2K
A How to derive the sampling distribution of some statistics
Replies
3
Views
1K
MHB Greatest probability - Expected value
Replies
3
Views
2K
I Is this conditional expectation identity true?
Replies
1
Views
2K
Chernoff Bounds using importance sampling identity
Replies
0
Views
2K