1. Random Variables A random variable is a variable whose value is determined by the outcome of a random experiment. Example: In a single die toss experiment, the sample space consists of six elements, 1, …, 6, denoted by  i, i=1,…,6. A random variable may be defined for this experiment: let the value of the random variable be equal to the value of the dice, i.e x(  i ) = i, i=1, 2, …, 6 The random variables discussed in our example could take on only a set of discrete numbers. Such random variables are know as discrete random variables (i.e. variables assume countable values). Random variables of another type, know as continuous random variables, may take on values anywhere within a continuous ranges.

2. Binomial Distribution In many Geographic studies, we often face a situation where we deal with a random variable that only takes two values, zero-one, yes-no, presence-absence, over a given period of time. Since there are only two possible outcomes, knowing the probability of one knows the probability of the other. P(1)=p P(0)=1-p=q If the random experiment is conducted n times, then the probability for the event to happen x times follow binomial distribution: Where n! the factorial of n. e.g. 5!=5*4*3*2*1=120.

3. For example, the presence-absence of drought in a year directly influence the profit of agriculture due to irrigation costs added in a dry year. Suppose a geographer is hired to do risk analysis for an Ag. Company whether a piece of land is profitable for agriculture. Past experience shows that irrigation can be afforded only one year in five. According to weather record, 4 out of the last 25 years suffered from drought in the area. Let 1 denote drought presence, and 0 denote drought absence, then P(1)=4/25=0.16, so P(0)=1-0.16=0.84. For 5 years, there are six possibilities of drought occurrence: 0, 1, 2, 3, 4, 5. Binomial Distribution Example

4. Drought occurrence probability in 5 years (probability mass function): The probability of profitable agriculture is summation of probabilities of no drought and one drought in five years, i.e. 0.418+0.398=0.816 This the risk is 18.4% in five years.

5. Poisson Distribution A discrete random variable is said to follow Poisson Distribution if its probability mass function is Where x = 0, 1, 2, …, and is the mean.

6. Poisson Distribution Example Suppose a geographer is assessing the risk of summer wheat yields to devastating hailstorms in a particular geographic location. Weather records show that in the past 35 years show that 10 years with no hailstorm, 13 years with one hailstorm, 8 years with 2 hailstorms, 3 years with 3 hailstorms, and 1 year with 1 hailstorm. Assume the occurrence of hailstorm is independent of past or future occurrences and can be considered random. Then the number of hailstorms happening in any given year follows Poisson distribution. In the above example, there are 42 hailstorms in 35 years, thus the mean number of hailstorms in a year is 1.2, then

7. Normal Distribution A continuous random variable is said to be normally distributed if its pdf is Where ( ,  ) are the distribution parameters x f(x) 

8. What Does the Mean Tell Us? For a random variable that follows normal distribution ( ,  ), f(x) x 11 22 The mean value tells us where the value x is concentrated most.

9. f(x) x  The variance tell how the value is spread. The larger the variance, the more even the value spreads over a large range. Is this good or bad? 11 22  2 >  1 What Does the Variance Tell Us?

10. f(x) x  x Does the variance change here? Why?

11. Prob -3 -2 -1 0 1 2 3 68.3% 95.5% 99.7% Standard Normal Distribution

12. Hypothesis Testing One application for the probability distribution is hypothesis testing, which is a standard statistical analysis for “difference” or “effect”. For example, before a new drug is put into market, FDA requires a detailed statistical analysis report on how effective the new drug is. This often requires a lot of random experiments. What they do is they usually ask a group of volunteers to test the new drug, and in the meantime, they have another group who may not take anything or a traditional drug for the same purpose. Then they test how effective the new drug is. The way they do this analysis is based on two statements: Statement 1: The new drug is not effective Statement 2: The new drug is effective These two statement is mutually exclusive, meaning negation of statement 1 naturally goes to statement 2. These two statements are hypotheses. The experimental results from the volunteers will be used to test which statement is acceptable, or we call it hypothesis testing.

13. Prob  -3   -2   -    +   +2   +3  68.3% 95.5% 99.7% Hypothesis Testing Null Hypothesis (H 0 ): no effect or no difference Alternative Hypothesis(H 1 ): The null hypothesis is given on which a probability distribution will be developed and its probability is then used to test the hypotheses. However, the decision made based on the statistics is not always correct. We make mistakes. Types of error we have. Type I: H 0 is true,but is rejected Type II: H 0 is false, but is not rejected

14. Student t distribution Probability Density Function: Where k is degrees of freedom Mean: 0 Variance: k/(k-2), k>2

15. Exponential Distribution Probability density function Mean: 1/λ Variance: 1/ λ 2

16. Chi-Square distribution Probability Density Function: Where k is degrees of freedom, and x≥0 Mean: k Variance: 2k

17. F distribution Probability Density function: Where U 1 and U 2 are chi-square distribution with d 1 and d 2 degrees of freedom, respectively