1 / 9

Chapter 10

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.

ahmed-vang
Download Presentation

Chapter 10

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 ≥ 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

  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 ≥ 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

  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 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

  9. 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 < ∞

More Related