Mathematical Statistics Concepts Probability laws Binomial distributions

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

2. Probability laws • the probability thateitherA or B occurs(union) • the probability thatbothA and B occur(intersection) • thejointprobability of A and B • theconditionalprobability of B given A

3. Statistical Independence • If events A and B are independent, then

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

5. Probabilities Marginal Probability: Pr(Genotype= A1A1) = 0.16 Joint Probability: Pr(Genotype= A1A1, height ≤ 50) = 0.10 Conditional Probability:

6. If X is statistically independent of Y, then their joint probability is equal to the product of the marginal probabilities of X and Y Statistical Independence

7. 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

8. 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: Binomial Probability Function Probability Function

9. 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

10. 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

11. Mean and variance of linear functions • Multiplying a random variable by a constant Adding two random variables Xand 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

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

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