Biostatistics for Animal Science

30 Biostatistics for Animal Science
y 1 2 3 4 5
Frequency 1 2 4 2 1
P(y)
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Check if the table shows a correct probability distribution. What is the probability that y is
greater than three, P(y > 3)?
1) 0 ≤ P(y) ≤ 1 ⇒ OK
2) Σi P(yi) = 1 ⇒ OK
The cumulative frequency of y = 3 is 7.
F(3) = P(y ≤ 3) = P(1) + P(2) + P(3) = ( 1 /10) + ( 2 /10) + ( 4 /10) = ( 7 /10)
P(y > 3) = P(4) + P(5) = ( 2 /10) + ( 1 /10) = ( 3 /10)
P(y > 3) = 1 – P(y ≤ 3) = 1 – ( 7 /10) = ( 3 /10)
Expectation:
E(y) = µ = Σi yi P(yi) = (1) ( 1 /10) + (2) ( 2 /10) + (3) ( 4 /10) + (4) ( 2 /10) + (5) ( 1 /10) = ( 30 /10) = 3
Variance:
Var(y) = σ 2 = E{[y – E(y)] 2 } = Σi P(yi) [yi – E(y)] 2 =
( 1 /10) (1 – 3) 2 + ( 2 /10) (2 – 3) 2 + ( 4 /10) (3 – 3) 2 +( 2 /10) (4 – 3) 2 + ( 1 /10) (5 – 3) 2 = 1.2
3.2.2 Bernoulli Distribution
Consider a random variable that can take only two values, for example Yes and No, or 0
and 1. Such a variable is called a binary or Bernoulli variable. For example, let a variable y
be the incidence of some illness. Then the variable takes the values:
yi = 1 if an animal is ill
yi = 0 if an animal is not ill
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The probability distribution of y has the Bernoulli distribution:
p
Here, q = 1 – p
Thus,
y 1−
y
( y)
= p q for y = 0,1
P(yi = 1) = p
P(yi = 0) = q
The expectation and variance of a Bernoulli variable are:
E(y) = µ = p and σ 2 = Var(y) = σ 2 = pq
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