Probability Distributions.

Transcription

1 Probability Distributions

2 Probability Measuring Discrete Outcomes Plotting probabilities for discrete outcomes: NOTE: Area within histogram equal to Number of Heads P (at least 1 head) = = 0.75

3 Discrete Probability Distributions How many times we expect to flip a coin until its a head? How many times we expect to roll a die until we get a 1? These questions are answered using probability theory: First, we define the probability of single each event, using the Bernoulli distribution. Then, we combine independent events using the axioms of probability theory:

4 Discrete Probability Distributions

5 The Bernoulli Distribution A Bernoulli random variable has two possible outcomes. We label one outcome a success (1) and the other outcome a failure (0). NOTE: success does need not be something positive. In this case, we can define a success as a head OR a tail. Suppose we observe ten trials: P (success) = P (failures) =

6 Bernoulli Probability Distribution X ~ Bernoulli (p) Bernoulli Random Variable Only one parameter: the probability of success Simplest experiment has only two possible outcomes: YES (1) and NO (0) P (having boy) = 1 P (having a girl)

7 Binomial Probability Distribution X ~ Bin (n, p) Binomial Random Variable Extension of Bernoulli Distribution Many Trials Two parameters: n and p P(X) = Probability of X successes in n trials

8 Binomial Probability Distribution Extension of Bernouille Process (with n > 1) We calculate the probability as the following product: P(X) = (# of scenarios) * P(single scenario) For Example: 3 Trials (3 Coin Flips) P(Trial 1) * P (Trial 2) * P(Trial 3) How many branches does the probability tree have?

9 Binomial Probability Distribution 3 Trials (3 Coin Flips) P(0 successes) = P (3 failures) = (0.5) * (0.5) * (0.5) = # scenarios = Only 1 way to have 3 tails: P(3 successes) = P (0 failures) = (0.5) * (0.5) * (0.5) = # scenarios = Only 1 way to have 3 heads:

10 Binomial Probability Distribution P(2 successes & 1 failure) = (0.5) * (0.5) * (0.5) = # scenarios? = = P(1 success & 2 failures) = (0.5) * (0.5) * (0.5) = # scenarios? = = 0.375

11 Binomial Probability Distribution (# of scenarios): Calculated with permutations (2 successes & 1 failure) = 3 paths n = 3 trials X = success (heads) (3!) / ( 2! * 1!) = 3 * 2 * 1 / 2 * 1 * 1 = 6 / 2 = 3 (3 successes) = 1 path 3! / 3! = 3 * 2 * 1 / 3 * 2 * 1 = 6 / 6 = 1

12 Binomial Probability Distributions n = 25, p = 0.5 n = 25, p = 0.8

13 Poisson Probability Distribution X ~ Poisson (Lambda) Poisson Random Variable Number of events recorded in a sample of fixed area or time Usually applied to rare events (e.g., getting hit by lightning) A single parameter: Lambda Lambda = Mean = Variance NOTE: X = number of successes

14 Continuous Probability Distributions

15 Central Limit Theorem In probability theory, the central limit theorem states that the mean of a sufficiently large number of independent random variables, each with a finite mean and variance, will be normally distributed

16 Z-score Normal Distribution X ~ N (µ, σ) Data Data + 5 Data * 5 Every Normal Distribution can be described using only two parameters: Mean and S.D. Z score quantifies number of SDs a value is from the mean

17 Example: Snail Species Probability that a given snail would be a given size, if - in fact - it belongs to a given population Large species Small species Mean = 12mm, STD = 2 Mean = 4mm, STD = 1 Size Range: Size Range: (12 4) to (12 + 4) (4 2) to (4 + 2) 96% from 8 to 16 96% from 2 to 6 Z = (7 12) / 2 Z = -5 / 2 Z = -2.5 Find a snail = 7mm long. What species is it? Z = (7 4) / 1 Z = 3 / 1 Z = +3

18 Lognormal Distribution Log X ~ N (µ, σ) A probability distribution of a random variable whose logarithm is normally distributed. Like Normal Distribution, described using two parameters: Mean and S.D. (Limpert et al. 2001)

19 Lognormal Distribution Normal Lognormal Frequency Cumulative Frequency Cumulative

20 Exponential Distributions It describes the time between events in a Poisson process (i.e. a process in which events occur continuously and independently at a constant average rate) Like Poisson Distribution, it is described using only one parameter: Lambda (sometimes called Beta) X ~ Exp (Lambda) Frequency Cumulative

21 Exponential Distributions Increase: Modelling Human Population Growth Current projections show a continued human population increase, with the population reaching between 7.5 and 10.5 billion by the year 2050

22 Exponential Distributions Decrease: Light extinction in the water column Iz Io * exp^ kz Units of k? 1 / m

23 Exponential Distributions Asymmetrical Long right- hand tails Distributions skewed to the right

24 Summary Probability Distributions Frequency distributions summarize probabilities of different outcomes (processes/parameters) (e.g., both numerical and categorical variables) Describe distributions using one or more parameters (NOTE: Know lecture examples) Use frequency distributions to predict probability of discrete and continuous variables (outcomes) These probabilities yield p values for all possible test statistics, given the degrees of freedom

25 Definition (Non)Parametric Parametric statistics assume that data come from a specific probability distribution (a normal distribution) and make inferences about parameters of the distribution. Non-parametric statistics involves: - distribution free techniques do not rely on data belonging to a particular distribution. For instance, randomization tests, whereby observations are shuffled. - non-parametric statistics whose interpretation does not depend on fitting any parameterized distribution. For instance, statistics based on ranks of observations are in the core of many non-parametric approaches.

26 Parametric Statistics Benefits and Costs: - Parametric methods make more assumptions than nonparametric methods. If the extra assumptions are correct, parametric methods have more statistical power (produce more accurate and precise estimates.) - However, if those assumptions are incorrect, parametric methods can be very misleading. They can cause false positives (type I errors). Thus, they are often not considered robust.

27 Parametric Statistics Suggested Approach: - Use parametric tests whenever possible. -Take care to examine diagnostic statistics and to determine if extra assumptions are met. - Perform the matching non-parametric test and compare results. What causes disagreements? - Note: Because parametric statistics require a probability distribution, they are not distribution-free.

28 Exploring Assumptions Parametric tests based on the normal distribution assume: Independent Observations Interval or Ratio Data (not binomial / nominal) Normally Distributed Sampling Distribution Residuals of Tests Many tests: Homogeneity of Variances