Chapter 10

The Normal and t Distributions

The Normal Distribution

• A random variable Z (-∞ ∞) is said

to have a standard normal distribution if its probability distribution is of the form:

The area under p(Z) is equal to 1

Z has and

ZZZp ,

2

1exp)2/1()( 2

0z 1z

The Normal Distribution

Find α such that Pr (Z ≥ Zc) = α

Find Zc such that Pr (Z ≥ Zc) = αα is a specific amount of probability and Zc

is the critical value of Z that bounds α probability on the right-hand tailTable A.1 for a given probability we search for Z value

Other Normal Distributions

• Random variable X (-∞ ∞) is said to have a normal distribution if its probability distribution is of the form:

where b>0 and a can be any value.

and

X

a

b

X

bXp ,

2

1exp

2

1)(

2

ax bx

Other Normal Distributions

• Any transformation can be thought of as a transformation of the standard normal distribution

X

Xkk

X

kk bZaX

Other Normal Distributions

• α=Pr(X ≥ Xk)= Pr(Z ≥ Zk), where

• X has a normal distribution with μ=5 and σ=2Pr(X ≥ 6) ?

X has a normal distribution with μ=5 and σ=2

X

Xkk

X

The t Distribution

• The equation of the probability density function p(t) is quite complex:p(t) = f (t; df), -∞< t <∞

• t has and when df>2• Probability problems:

Find α such that Pr(t ≥ t*) =αTable A.2 can be used to find probabilitydf=5, Pr(t ≥ 1.5) = 0.097 and Pr(t ≥ 2.5) = 0.027

ot 2

df

dft

The Chi-Square Distribution

• When we have d independent random variables z1, z2 , z3, . . . Zd , each having a standard normal distribution.

• We can define a new random variable

χ2 = , df=d

Figure 10.8 page 222

χ2 has μ = d and σ =

Find (χ2 )c such that Pr(χ2 ≥ (χ2)c) =α

Table A.4 df =10 and α=0.10 then

χ2 ≥ (χ2) c =15.99

d

j jZ1

2)(

d2

The F Distribution

• Suppose we have two independent random variables χ2

n and χ2d having chi-square

distributions with n and d degrees of freedom• A new random variable F can be defined as:

• This random variable has a distribution with n and d degrees of freedom

• 0 ≤ F < ∞

d

nF

d

n

/)(

/)(2

2