# Bernoulli Trial help

• Dec 17th 2011, 06:00 AM
skyhawk714
Bernoulli Trial help
hey guys i have my final coming up and am really struggling. anyone care to help with this problem?

A probability experiment consists of conducting independent Bernoulli trials with probability of success 0.32 until the first success occurs. The expected number of failures preceding the first success has expected value
• Dec 17th 2011, 06:47 AM
chisigma
Re: Bernoulli Trial help
Quote:

Originally Posted by skyhawk714
hey guys i have my final coming up and am really struggling. anyone care to help with this problem?

A probability experiment consists of conducting independent Bernoulli trials with probability of success 0.32 until the first success occurs. The expected number of failures preceding the first success has expected value

Often it does happen that in [apparently...] difficult probability problems the solution is surprisingly simple!... in Bernoulli trials the probability to first succes at the n-th trial is...

$\displaystyle p_{n}=p\ (1-p)^{n-1}$ (1)

From (1) You derive that...

$\displaystyle E\{n\}= \sum_{n=1}^{\infty} n\ (1-p)^{n-1}\ p= p\ \sum_{n=1}^{\infty} n\ (1-p)^{n-1}$ (2)

Now You set $\displaystyle 1-p=x$ and You obtain...

$\displaystyle E\{n\}= (1-x)\ \sum_{n=1}^{\infty} n\ x^{n-1} = (1-x)\ \frac{d}{d x} \frac{1}{1-x}= \frac{1}{1-x}=\frac{1}{p}$ (3)

http://www.sv-luka.org/ikone/ikone180a.jpg

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$