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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.

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Chapter 10

Chapter 10

The Normal and t Distributions


The normal distribution
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 distribution1
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
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 distributions1
Other Normal Distributions

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


Other normal distributions2
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 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
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
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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