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Let's say I'm tossing a coin until I get the first heads. I repeat that experiment n times. Then I'll take the maximum value of tries that were needed to get heads in that experiment (out of the n results obtained). I was expecting that value to be in average $\log_2 n$, but I ran a simulation on matlab and the results didn't quite match it.

The results were always around 1.3 greater than that even using millions of sample sizes. The weird thing is that the histogram of the number of tries needed to get heads for the first time is spot on, so I know that the problem isn't on my code, but on my math. The result link shows the plots I got from running 10000 times the experiment to get the maximum value of tries considering that each time $2^{14}$ values were recorded.

I expected the mean of the max to be 14, but I got it to be on average 15.3 with a standard deviation of 1.89. My question is, how can I get mathematically to this average and standard deviation, and how can I find the expression to plot that curve?

Also, please correct me if I'm wrong but the reason this doesn't follow a normal distribution is due to the fact that the result is limited from [1,inf[, so it is truncated, right?

user
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Israel Solha
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  • Geometric distribution? – Peter Foreman Apr 12 '19 at 19:47
  • Yes, the first part is a geometric distribution. But then I’m looking at the the average of the greatest values of that distribution with n samples. The geometric distribution works just fine, my problem is in the next step. – Israel Solha Apr 12 '19 at 19:58
  • Welcome to MSE. Please edit and use MathJax to properly format numbers and math expressions. Also, would you kindly split the text into several paragraphs (and perhaps with some formatting to highlight your points)? It would be easier to read. – Lee David Chung Lin Apr 12 '19 at 21:01
  • Hint: Observe that $\frac12+\frac\gamma{\log 2}\approx1.33275$. Compare this value with the deviation from the result, you have expected. – user Apr 17 '19 at 20:59

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