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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.

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Chapter 10

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  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 < ∞

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