# Math Help - How many coin flips until first tail/head

1. ## How many coin flips until first tail/head

1. How many coin flips are expected for the first tail?
2. $X$ is number of coin flips before first tail. What is dispersion $D(X)$?

1. I know this one ... $\sum_{i=0}^\infty \frac{1}{2^i} = 2$
2. ... but not this one: dispersion is defined as $D(X)=E(X-E(X))^2=E(X^2)-(E(X))^2=???$ What is $E(X^2)$? Does before suggest that $E(X)=2-1$?

The answer to the 2nd question is $2$.

2. This may not help you too much (assuming you intend to run through the math) but at least in my book, they do this dirty derivation and then say "just remember this"

with "this" being:
for a geometric distribution (one where you repeat until success with probably p of success each trial), the "dispersion" (we know it as variance) is
$Var(X)=\frac{1-p}{p^2}=\frac{1-.5}{.5^2}=2$

3. Originally Posted by courteous
1. I know this one ... $\sum_{i=0}^\infty \frac{1}{2^i} = 2$
2. ... but not this one: dispersion is defined as $D(X)=E(X-E(X))^2=E(X^2)-(E(X))^2=???$ What is $E(X^2)$? Does before suggest that $E(X)=2-1$?
The answer to the 2nd question is $2$.
You will find the derivation of the formula in post #2 in most standard textbooks on mathematical statistics. I'm sure it can be found using Google too. And I'm sure the derivation has been done several times already at MHF (search the threads).