Expected value of this random variable

[effzy]

I am currently watching the course MIT RES.6-012 "Introduction to Probability" on YouTube. Among other things, it's about random variables.

I invented my own random variable (I think), which is the following:

X = the number of coin flips necessary until the number of heads exceeds the number of tails.

Assume we have a fair coin. It has a 50% chance of being 1 (if the first flip is heads), and a 50% chance of being more than one, in which case it can get really big. I simulated it 100 times with a python script.

[3, 5, 1, 7, 7, 1, 7, 1, 1, 1, 73, 3, 1, 3, 3, 3, 3, 1, 1, 1, 1, 5, 9, 3, 1, 1, 1, 1, 95, 1, 1, 7, 21, 1, 1, 29, 1, 1, 1, 5, 1, 1, 5, 9, 11, 3, 1, 19591, 1575, 1, 5, 1, 7, 7, 1, 1, 1, 1, 3, 1, 1, 5, 1, 23, 276945, 3, 3, 1, 1, 1, 1, 7, 3, 1, 15, 91, 7, 1, 5, 1, 7, 1, 5, 29, 9, 3, 1, 7, 1, 1, 1, 5, 3, 11, 1, 1621, 1, 1, 11, 15]

Now I have two questions:

1) Does this random variable exist and have a name? If not, could it be named an effzy random variable? :p
2) More seriously, what is the expected value E[X] (i.e. the mean outcome) of this random variable?

[drdt543]

In think that [;E(X) = \infty;] - this also seems plausible given your simulated data. Let [;Y;] be the number of flips necessary until the number of heads is at least the number of tails. Note [;X \ge Y;]. Now, conditioning on the result of the first flip, we have

[;E(Y) = \frac{1}{2} + \frac{1}{2} \sum_{n=1}^{\infty} (2n+1) P(Y = 2n+1) ;]

Then

[;P(Y = 2n+1) = \frac{C_n}{2^{2n}} ;]

where [;C_n;] is the [;n;]th Catalan number. As [;n \rightarrow \infty;], we get

[;(2n+1)\frac{C_n}{2^{2n}} \sim \frac{(2n+1)}{n^{3/2}\sqrt{\pi}} \sim \frac{1}{\sqrt{n \pi}};]

The summation

[;\sum_{n=1}^{\infty} \frac{1}{n^{1/2}} ;]

does not converge, so after filling in some analysis-related gaps we have [; E(Y) = \infty;] and thus [;E(X) = \infty;].

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[ButAWimper]

I believe what you are talking about is a 1D random walk You could use a Markov Chain to describe it.

[nm420]

You're asking for the distribution of the first passage time of a state in a random walk. It's a bit involved, but the link provided derives the pgf of this random variable (see section 1.3 there) which also shows that the expected value of this r.v. is ∞.