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### 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

The Normal Distribution

Find α such that Pr (Z ≥ Zc) = α

Find Zc such that Pr (Z ≥ Zc) = α

α is a specific amount of probability and Zc is the critical value of Z that bounds α probability on the right-hand tail

Table 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

Other Normal Distributions

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

Other Normal Distributions

- α=Pr(X ≥ Xk)= Pr(Z ≥Zk), where
- X has a normal distribution with μ=5 and σ=2

Pr(X≥ 6) ?

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

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*) =α

Table A.2 can be used to find probability

df=5, Pr(t ≥ 1.5) = 0.097 and Pr(t ≥ 2.5) = 0.027

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

The F Distribution

- Suppose we have two independent random variables χ2n and χ2dhaving 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 < ∞

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