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Mathematical Statistics Concepts Probability laws Binomial distributions

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)

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Mathematical Statistics Concepts Probability laws Binomial distributions

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  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)

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