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

2. 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

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 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:

6. 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

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. 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

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 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

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)