基本概率分布Basic Concept of Probability Distributions 3: Geometric Distribution
PMF
Suppose that independent trials, each having a probability $p$, $0 < p < 1$, of being a success, are performed until a success occurs. If we let $X$ equal the number of failures required, then the geometric distribution mass function is $$f(x; p) =\Pr(X=x) = (1-p)^{x}p$$ for $x=0, 1, 2, \cdots$.
Proof:
$$ \begin{align*} \sum_{x=0}^{\infty}f(x; p) &= \sum_{x=0}^{\infty}(1-p)^{x}p\\ &= p\sum_{x=0}^{\infty}(1-p)^{x}\\ & = p\cdot {1\over 1-(1-p)}\\ & = 1 \end{align*} $$
Mean
The expected value is $$\mu = E[X] = {1-p\over p}$$
Proof:
Firstly, we know that $$\sum_{x=0}^{\infty}p^x = {1\over 1-p}$$ where $0 < p < 1$. Thus $$ \begin{align*} {d\over dp}\sum_{x=0}^{\infty}p^x &= \sum_{x=1}^{\infty}xp^{x-1}\\ &= {1\over(1-p)^2} \end{align*} $$ The expected value is $$ \begin{align*} E[X] &= \sum_{x=0}^{\infty}x(1-p)^{x}p\\ &=p(1-p)\sum_{x=1}^{\infty}x(1-p)^{x-1}\\ &= p(1-p){1\over(1-(1-p))^2}\\ &= {1-p\over p} \end{align*} $$
Variance
The variance is $$\sigma^2 = \mbox{Var}(X) = {1-p\over p^2}$$
Proof:
$$ \begin{align*} E\left[X^2\right] &=\sum_{x=0}^{\infty}x^2(1-p)^{x}p\\ &= (1-p)\sum_{x=1}^{\infty}x^2(1-p)^{x-1}p \end{align*} $$ Rewrite the right hand summation as $$ \begin{align*} \sum_{x=1}^{\infty} x^2(1-p)^{x-1}p&= \sum_{x=1}^{\infty} (x-1+1)^2(1-p)^{x-1}p\\ &= \sum_{x=1}^{\infty} (x-1)^2(1-p)^{x-1}p + \sum_{x=1}^{\infty} 2(x-1)(1-p)^{x-1}p + \sum_{x=1}^{\infty} (1-p)^{x-1}p\\ &= E\left[X^2\right] + 2E[X] + 1\\ &= E\left[X^2\right] + {2-p\over p} \end{align*} $$ Thus $$E\left[X^2\right] = (1-p)E\left[X^2\right] + {(1-p)(2-p) \over p}$$ That is $$E\left[X^2\right]= {(1-p)(2-p)\over p^2}$$ So the variance is $$ \begin{align*} \mbox{Var}(X) &= E\left[X^2\right] - E[X]^2\\ &= {(1-p)(2-p)\over p^2} - {(1-p)^2\over p^2}\\ &= {1-p\over p^2} \end{align*} $$
Examples
1. Let $X$ be geometrically distributed with probability parameter $p={1\over2}$. Determine the expected value $\mu$, the standard deviation $\sigma$, and the probability $P\left(|X-\mu| \geq 2\sigma\right)$. Compare with Chebyshev's Inequality.
Solution:
The geometric distribution mass function is $$f(x; p) = (1-p)^{x}p,\ x=0, 1, 2, \cdots$$ The expected value is $$\mu = {1-p\over p} = 1$$ The standard deviation is $$\sigma = \sqrt{1-p\over p^2} = 1.414214$$ The probability that $X$ takes a value more than two standard deviations from $\mu$ is $$P\left(|X-1| \geq 2.828428\right) = P(X\geq 4) = 0.0625$$ R code:
1 - sum(dgeom(c(0:3), 1/2)) # [1] 0.0625
Chebyshev's Inequality gives the weaker estimation $$P\left(|X - \mu| \geq 2\sigma\right) \leq {1\over4} = 0.25$$
2. A die is thrown until one gets a 6. Let $V$ be the number of throws used. What is the expected value of $V$? What is the variance of $V$?
Solution:
The PMF of geometric distribution is $$f(x; p) = (1-p)^xp,\ = 0, 1, 2, \cdots$$ where $p = {1\over 6}$. Let $X = V-1$, so the expected value of $V$ is $$ \begin{align*} E[V] &= E[X+1]\\ &= E[X] + 1\\ &= {1-p\over p} + 1\\ &= {1-{1\over6} \over {1\over6}} + 1\\ &= 6 \end{align*} $$ The variance of $V$ is $$ \begin{align*} \mbox{Var}(V) &= \mbox{Var}(X+1)\\ &= \mbox{Var}(X)\\ &= {1-p\over p^2}\\ &= {1-{1\over 6} \over \left({1\over6}\right)^2}\\ &= 30 \end{align*} $$ Note that this is another form of the geometric distribution which is so-called the shifted geometric distribution (i.e. $X$ equals to the number of trials required). By the above process we can see that the expected value of the shifted geometric distribution is $$\mu = {1\over p}$$ and the variance of the shifted geometric distribution is $$\sigma^2 = {1-p\over p^2}$$
3. Assume $W$ is geometrically distributed with probability parameter $p$. What is $P(W < n)$?
Solution:
$$ \begin{align*} P(W < n) &= 1 - P(W \geq n)\\ &= 1-(1-p)^n \end{align*} $$
4. In order to test whether a given die is fair, it is thrown until a 6 appears, and the number $n$ of throws is counted. How great should $n$ be before we can reject the null hypothesis $$H_0: \mbox{the die is fair}$$ against the alternative hypothesis $$H_1: \mbox{the probability of having a 6 is less than 1/6}$$ at significance level $5\%$?
Solution:
The probability of having to use at least $n$ throws given $H_0$ (i.e. the significance probability) is $$P = \left(1 - {1\over 6}\right) ^n$$ We will reject $H_0$ if $P < 0.05$. R code:
n = 1
while (n > 0){
+ p = (5/6) ^ n
+ if (p < 0.05) break
+ n = n + 1
+ }
n
# [1] 17
That is, we have to reject $H_0$ if $n$ is at least 17.
Reference
- Ross, S. (2010). A First Course in Probability (8th Edition). Chapter 4. Pearson. ISBN: 978-0-13-603313-4.
- Brink, D. (2010). Essentials of Statistics: Exercises. Chapter 5 & 10. ISBN: 978-87-7681-409-0.