1. Chapter 10 The Normal and t Distributions

2. 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

3. The Normal Distribution Find α such that Pr (Z ≥ Z c ) = α Find Z c such that Pr (Z ≥ Z c ) = α α is a specific amount of probability and Z c 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

4. 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

5. Other Normal Distributions Any transformation can be thought of as a transformation of the standard normal distribution

6. Other Normal Distributions α=Pr(X ≥ X k )= Pr(Z ≥ Z k ), 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

7. 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

8. The Chi-Square Distribution When we have d independent random variables z 1, z 2, z 3,... Z d, each having a standard normal distribution. We can define a new random variable χ 2 =, df=d Figure 10.8 page 222 χ 2 has μ = d and σ = Find ( χ 2 ) c such that Pr(χ 2 ≥ (χ 2) c ) =α Table A.4 df =10 and α=0.10 then χ 2 ≥ (χ 2 ) c =15.99

9. The F Distribution Suppose we have two independent random variables χ 2 n and χ 2 d 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 < ∞