Other random number distributions

Discrete probability distribution

Geometric Distribution

• Examples:
  • Throw a die until you get ``2'', how many times do you have to throw the die?
    
    
  • There is a probability (p=0.1) that a population goes extinct (by flood) every year. What is the expected time until extinction of a population?
• Different from Binomial R.V. which is the number of successes in a fixed number of n trials.
• Probability of success: p
  R.V. X = number of trials to get the first success.
  Pr[k-1 unsuccessful tries] x Pr[success]
  $$\begin{eqnarray*} Prob[X=k] &=& (1-p)^{k-1} p \\ E[X] &=& 1/p \\ Var[X] &=& (1-p)/p^2 \end{eqnarray*}$$
  
  
  

Negative Binomial Distribution

• Simple extension of Geometric distribution
• Example:
  • How many times do you have to throw a die until you get ``2'' r times.
  • If a flower requires 5 pollinator visits to export all of the pollen, how long does it take to export all pollen, given that given that a probability of a pollinator visit is 0.1 per hour.
• The probability of ``success'' is p. We want to describe the waiting time (X) to get r success.

  Prob[X=7]: Probability that you get the 3rd success (r=3) at the 7-th trial:
  
  

  $$\begin{eqnarray*} Prob[X=k] &=& {k-1 \choose r-1} p^{r-1} (1-p)^{k-r} p = {k-1 ... ...1} p^r (1-p)^{k-r} \\ E[X] &=& r/p\\ Var[X] &=& r (1-p)/p^2 \end{eqnarray*}$$
  
  

  

Poisson Distribution

• Examples
  1. number of a defective product from a factory per year
  2. number of typo's per page
  3. number of death due to moose stomping per year
  4. number of mutations per genome in one generation
• counts of rare events
• Poisson distribution can be used to approximate a binomial dstribution when number of trials (n) is large and p is small
• Probability mass function
  
  $\parbox{3in}{ \begin{displaymath}Pr[X=k] = \frac{e^{-\lambda} \lamb... ...ambda \end{displaymath} \begin{displaymath}Var[X] = \lambda \end{displaymath} }$

  

Exercises

1. A fire station receives 730 alarms per year. What is the probability that on a given day, there will be no alarms? How about 4 alarms on a given day?
   
   
   
2. A wolf pack goes hunting for a carribou. If the probability of catching a carribou is 0.2 per day, how many days on average does it take for them to get a caribou?
3. Let's say that a tadpole requires to catch 10 shrimps before it metamorphs. If they catch 1 shrimp every other day on average, how many days does it take for a tadpole to metamorph?
4. In a population with meiotic drive (probability of a female offspring is p), what is the probability that you'll see no male in 5 offsprings?
5. If there are $3 \times 10^9$ basepairs in the human genome and the mutation rate per generation per basepair is $10^{-9}$, what is the mean number of new mutations per genome that a child will have?

Continuous probability distribution

In a discrete random variable, you can calculate the probability of X is at a particular value (e.g, X=3, X=0 etc) because there are discrete set of outcomes.

In a continous random variable, there are infinite number of points, so the probability that something happens at a particular point (e.g. Prob[X = 1.4142]) is nearly zero. But you can calculate the probability that something happens within an interval.

Probability of an event happening within an interval between a and b:

$$Pr[a \le X \le b] = \int^b_a f(x) dx$$

f(x) is called probability density function, and it's shape characterize the distribution of random variable.

Exponential Distribution

• Continuous version of geometric distribution.
  Time between events that happen at a constant average rate.
  Time for a continuous process to change the state.
• Examples:
  1. The time until a light bulb burns out
  2. The time until your next car accident.
  3. Distance between roadkill
• probability density function
  $$\begin{eqnarray*} f(x) &=& \theta e ^ {- \theta x} \\ E[X] &=& 1/\theta \\ Var[X] &=& 1/\theta^2 \end{eqnarray*}$$
  
  
  
• Cumulative distribution function: Probability that a R.V. X is less than a value x
  
  $$F(x) = \int_{-\infty}^x f(x) dx = 1 - e^{-\theta x}$$
  
  
  
• Side note: Negative binominal distribution was a simple extension of geometric distribution. gamma distribution is a continuous version of negative binominal distribution.

Normal distribution

• Familiar distribution
  Measurement error
  A lot of small effects acting additively to determint the value
  e.g., blood pressure
  
• Probability density function
  
  $$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} \cdot exp\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right)$$