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Distributions Derived from the Normal

Distributions Derived from the Normal. Overview. Distributions derived from normal Chi-square distribution (Student) t distribution F distribution Sample mean and sample variance (for independent normal random variables). Gamma Function. The gamma function, (x), is defined by (1) = 1

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Distributions Derived from the Normal

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  1. Distributions Derived from the Normal

  2. Overview • Distributions derived from normal • Chi-square distribution • (Student) t distribution • F distribution • Sample mean and sample variance (for independent normal random variables)

  3. Gamma Function • The gamma function, (x), is defined by • (1) = 1 • (x+1) = x (x) • For positive integers n, (n) = (n-1)! • (1/2) = 

  4. Chi-Square Distribution • Definition. Suppose Z is a standard normal r.v. The distribution of U = Z2 is the chi-square distribution with 1 degree of freedom • Notation: U ~ 12 • Density:

  5. Chi-Square Distribution • Definition. Suppose U1,U2,…,Un are independent chi-square r.v.’s each with 1 degree of freedom. The distribution of V = U1 + U2 + … + Un is the chi-square distribution with n degrees of freedom • If U is chi-square with n degrees of freedom and V is chi-square with m degrees of freedom, then U + V is chi-square with n+m degrees of freedom

  6. Chi-Square Distribution • Suppose U is chi-square with n degrees of freedom • Notation: U ~ n2 • Density: • E(U) = n • Var(U) = 2n • U is gamma with  = n/2 and  = 1/2

  7. Chi-Square Distribution

  8. (Student) t distribution • Suppose Z ~ N(0,1) and U ~ n2 and Z and U are independent. • Then has a t distribution with n degrees of freedom • Density:

  9. (Student) t distribution • Note that the t distribution is symmetric about 0 • f(x) = f(-x) • As the degrees of freedom becomes large the t distribution converges (in distribution) to the standard normal

  10. Normal and t distributions

  11. F Distribution • Definition. Suppose U and V are independent chi-square distributions with respective degrees of freedom n and m. The the distribution of W = (U/n) / (V/m) is the F distribution with degrees of freedom m and n • Notation: W ~ F m,n

  12. Sample Mean • Suppose that X1, X2,..,Xn is an independent sequence of N(m,s2 ) random variables • The sample mean is • The expected value of the sample mean is m and its variance is s2/n • Moreover, the distribution of the sample mean is N (m,s2/n)

  13. Sample Variance • Suppose that X1, X2,..,Xn is an independent sequence of N(m,s2 ) random variables • The sample variance is • The mean of the sample variance is s2

  14. Sample Mean and Variance • Theorem. The random variables and are independent • Theorem.

  15. Sample Mean and Variance • Finding likelihood is an accurate estimate of a = .9 b = 1.1 n

  16. Sample Mean and Variance • Corollary. Suppose that X1, X2,..,Xn is an independent sequence of N(m,s2 ) random variables. Then

  17. Examples • Chapter 6, #3 • Chapter 6, #9 • Chapter 6, #10

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