1. Random Variables & Probability Distributions Outcomes of experiments are, in part, random E.g. Let X 7 be the gender of the 7 th randomly selected student. In this case, the sample space is S={M,F} Probability distributions used to understand, model, and predict outcomes of random experiments. Many useful distributions for describing random processes in environmental science & mgt.

2. Example: Hazardous Waste Hazardous Waste Depository: test wells-monitor groundwater for leaks. Aldicarb limit = 30 ppb Aldicarb occurs naturally (but concentration is variable). What is probability of exceeding limit even if no leak? (Prob measuring > 30 even if no leak?)

3. Evidence and Data “Natural” distribution of aldicarb: 500 readings from sites known to not be contaminated:

4. Evidence Cont’d Based on this distribution, we will assume these data are normally distributed with: Mean = 20 ppb Standard Deviation = 4 ppb

5. Definitions Random Variable: the unknown outcome of an experiment. The particular outcome is a realization of the random variable. E.g. (1) rain Tues., (2) aldicarb measurement r.v. takes diff. values each w/ diff. probs. Histogram: plot of the frequency of observation of a random variable over discrete intervals. Discrete vs. Continuous Random Variable

6. Frequency of Outcomes Probability Density (Mass) Function: Histogram of outcomes resulting from infinite # samples: (Prob = area under) For cont., bar width approaches 0 Cumulative Distribution Function: Probability that the r.v.  x. Examples on board: # Grizzly cubs per sow (1,2,1,2,2,2,2,3,1,2) Histogram vs. known prob. mass (.13,.70,.17) Natural aldicarb concentration Histogram (of data) vs. pdf N(20,4)

7. Known vs. Unknown Distributions True distribution may not be a known distribution (e.g. dist’n of student’s heights in this classroom) Often, knowing how a process works will point us to a particular (known) distribution Advantages of known distributions: Can usually be described by 1 or 2 parameters. Well studied, so most properties known Easy to ask questions like the aldicarb question.

8. Discrete Random Variables 1. Bernoulli: 2 outcomes: “success” (prob,= p) or “failure” (prob.= 1-p) 2. Binomial: Number of successes in n independent Bernoulli trials. 3. Multinomial: Extends Binomial to more than 2 outcomes. 4. Geometric: Number Bernoulli trials until first success. 5. Poisson: Counting r.v. (takes integer values). Number events that occur in given time interval.

9. Normal Random Variable 1. Normal: “Bell Shaped”, “Gaussian”. Symmetric. + and – values. 1. Central Limit Theorem: Sum or Avg. of several independent r.v.’s, result is normal (often used as justification for Normal). 2. “Standard Normal”: N(0,1). 3. Convert X~N(,) to Standard Normal (Z): Z=(X-)/

10. Continuous Random Variables 1. Uniform: every possible outcome equally likely (also a discrete r.v.) 2. Log-Normal: r.v. whose logarithm is normally distributed. 3. Gamma: Non-negative values. 4. Extreme Value: Maximum or minimum of many draws from some other distribution. 5. Exponential: Inter-arrival times, “memoryless”.   2 : Closely related to Normal. Non- negative. Skewed.

11. Answer Question: What is probability that measured aldicarb level  30 ppb, if no leak? Let X be a random variable describing the aldicarb level of a given test. P(X  30) = area under N(20,4) above 30 ppb.

12. Integrate Under N(20,4) Normal pdf: Draw on board…Ouch! Isn’t there another way?

13. 2 Ways to Answer 1. Ask S-Plus (nicely): P(X 30)=0.006. 2. Convert to N(0,1). 1. Standard Normal Z=(30-20)/4=2.5. Table gives Pr(0<Z<z): z.00.01.02.03 0.0.000.004.008.012 0.3.118.122.126.129 1.8.464.465.466 2.5.494

14. Answer Pr(X>30) when X~N(20,4) = Pr(Z>2.5) when Z~N(0,1) Pr(0<Z<2.5)=.494 Pr(-  <Z<0)=.500 So, Pr(Z>2.5)=1-.494-.5 =.006