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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)
Pr( A  B)  Pr( A )  Pr(B)  Pr( A  B)
•
•
the probability that both A and B occur (intersection)
the joint probability of A and B
Pr( A  B)  Pr( A,B)
•
the conditional probability of B given A
Pr( A,B)
Pr( B A ) 
Pr( A )
Statistical Independence
• If events A and B are independent, then
Pr( B A )  Pr( B)
Pr( A B)  Pr( A )
Pr( A,B)  Pr( A ) Pr( B)
Bayes’ Theorum (Bayes’ Rule)
•
•
Pr( A B ) 
Conditional probability
Bayes’ Theorum
Pr( A B) 
Pr( A, B )
Pr(B )
Pr(B A)Pr  A 
Pr(B)
.
Pr(A) is called the prior probability
Pr(A|B) is called the posterior probability
Pr( Aj B ) 
Pr(B Aj )Pr  Aj 
k
Pr(B A )Pr  A 
i 1
i
i
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
0.10
0.32
height > 75 cm
0.02
0.16
0.20
0.38
Marginal Prob.
0.16
0.48
0.36
1.00
Marginal Probability:
Pr(Genotype= A1A1) = 0.16
Joint Probability:
Pr(Genotype= A1A1, height ≤ 50) = 0.10
Conditional Probability:
Pr height  50 Genotype  A1A1 

Pr (height  50, Genotype  A1A1 ) 0.10

 0.625
Pr( Genotype  A1A1 )
0.16
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
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
Marginal Prob.
0.16
0.48
0.36
1.00
Plant Height
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:
n!
n 
y
y! (n  y )!
– Calculate the probability of that result:


n!
n y
Pr( y ) 
p y 1  p
y! (n  y )!

Probability Function
Binomial Distribution
Probability
Binomial Distribution (n=20, p=0.5)
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Number of successes
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)
E (c )  c
 0
2
c
•
Adding a constant (c) to a random variable Xi
E( X  c )  E( X )  c
the mean increases by the
value of the constant

the variance remains the same
2
X c

2
X
Mean and variance of linear functions
•
Multiplying a random variable by a constant
E(cX )  c[E( X )]
multiply the mean by the constant

multiply the variance by the
square of the constant
2
cX
c 
2
2
X
Adding two random variables X and Y
E( X  Y )  E( X )  E(Y )
mean of the sum is the sum
of the means
(2X Y )  VAR( X )  VAR(Y )  2COVXY
variance of the sum  the sum of the variances if the variables
are independent
Variance - definition
• The variance of variable X
V ( X )  E ( X i  X )2   E ( X i2 )  X2
• Usual formula
X


 
2
X
i
 X
n
2
2


X


 i  n
   X i2 
n




• Formula for frequency data (weighted)
V ( X )   X2   f i X i -  X2
Covariance - definition
• The covariance of variable X and variable Y
Cov ( X ,Y )  E ( X  X )(Y  Y )  E ( XY )  X Y
• Usual formula
X


XY 
i
 X  Yi  Y 
n

X i  Yi 

   Xi Yi 
 n
n


• Formula for frequency data (weighted)
Cov ( X , Y )   XY   f i X iYi -  X Y