PBG 650 Advanced Plant Breeding Mathematical Statistics Concepts Probability laws Binomial distributions Mean and Variance of Linear Functions

Probability laws the probability that either A or B occurs (union) the probability that both A and B occur (intersection) the joint probability of A and B the conditional probability of B given A

Statistical Independence If events A and B are independent, then

Bayes’ Theorum (Bayes’ Rule) Conditional probability Bayes’ Theorum                           . Pr(A) is called the prior probability Pr(A|B) is called the posterior probability

Probabilities Plant Height A1A1 A1A2 A2A2 Marginal Prob. height ≤ 50 cm 0.10 0.14 0.06 0.30 50 < height ≤ 75 0.04 0.18 0.32 height > 75 cm 0.02 0.16 0.20 0.38 0.48 0.36 1.00 Marginal Probability: Pr(Genotype= A1A1) = 0.16 Joint Probability: Pr(Genotype= A1A1, height ≤ 50) = 0.10 Conditional Probability:

Statistical Independence If X is statistically independent of Y, then their joint probability is equal to the product of the marginal probabilities of X and Y Plant Height A1A1 A1A2 A2A2 Marginal Prob. height ≤ 50 cm 0.0480 0.1440 0.1080 0.30 50 < height ≤ 75 0.0512 0.1536 0.1152 0.32 height > 75 cm 0.0608 0.1824 0.1368 0.38 0.16 0.48 0.36 1.00

Discrete probability distributions Let x be a discrete random variable that can take on a value Xi, where i = 1, 2, 3,… The probability distribution of x is described by specifying Pi = Pr(Xi) for every possible value of Xi 0 ≤ Pr(Xi) ≤ 1 for all values of Xi ΣiPi = 1 The expected value of X is E(X) = ΣiXiPr(Xi) =X

Binomial Probability Function A Bernoulli random variable can have a value of one or zero. The Pr(X=1) = p, which can be viewed as the probability of success. The Pr(X=0) is 1-p. A binomial distribution is derived from a series of independent Bernouli trials. Let n be the number of trials and y be the number of successes. Calculate the number of ways to obtain that result: Calculate the probability of that result: Probability Function

Binomial Distribution Average = np = 20*0.5 = 10 Variance = np(1-p) = 20*0.5*(1-0.5) = 5 For a normal distribution, the variance is independent of the mean For a binomial distribution, the variance changes with the mean

Mean and variance of linear functions Mean and variance of a constant (c) Adding a constant (c) to a random variable Xi the mean increases by the value of the constant the variance remains the same

Mean and variance of linear functions Multiplying a random variable by a constant Adding two random variables X and Y multiply the mean by the constant multiply the variance by the square of the constant mean of the sum is the sum of the means variance of the sum  the sum of the variances if the variables are independent

Variance - definition The variance of variable X Usual formula Formula for frequency data (weighted)

Covariance - definition The covariance of variable X and variable Y Usual formula Formula for frequency data (weighted)