Sampling Distributions

Transcription

1 Sampling Distributions Mathematics 47: Lecture 9 Dan Sloughter Furman University March 16, 2006 Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

2 Definition We call the probability distribution of a statistic T a sampling distribution. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

3 Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

4 Let X 1, X 2,..., X n be a random sample from a continuous distribution with probability density function f and (cumulative) distribution function F. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

5 Let X 1, X 2,..., X n be a random sample from a continuous distribution with probability density function f and (cumulative) distribution function F. Let Y = X (n). Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

6 Let X 1, X 2,..., X n be a random sample from a continuous distribution with probability density function f and (cumulative) distribution function F. Let Y = X (n). If G is the distribution function of Y, then G(y) = P(Y y) = P(X 1 y, X 2 y,..., X n y) n = P(X i y) = (F (y)) n. i=1 Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

7 Let X 1, X 2,..., X n be a random sample from a continuous distribution with probability density function f and (cumulative) distribution function F. Let Y = X (n). If G is the distribution function of Y, then G(y) = P(Y y) = P(X 1 y, X 2 y,..., X n y) n = P(X i y) = (F (y)) n. i=1 Hence, if g is the probability density function of Y, then g(y) = d dy G(y) = n(f (y))n 1 f (y) = nf (y)(f (y)) n 1. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

8 (cont d) Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

9 (cont d) Now let W = X (1). Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

10 (cont d) Now let W = X (1). If H is the the (cumulative) distribution function of W, then H(w) = P(W w) = 1 P(W > w) = 1 P(X 1 > w, X 2 > w,..., X n > w) n = 1 P(X i > w) = 1 (1 F (w)) n. i=1 Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

11 (cont d) Now let W = X (1). If H is the the (cumulative) distribution function of W, then H(w) = P(W w) = 1 P(W > w) = 1 P(X 1 > w, X 2 > w,..., X n > w) n = 1 P(X i > w) = 1 (1 F (w)) n. i=1 Hence, if h is the probability density function of W, then h(w) = d dy H(w) = n(1 F (w))n 1 ( f (w)) = nf (w)(1 F (w)) n 1. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

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13 Suppose X 1, X 2,..., X n is a random sample from a uniform distribution on (0, θ). Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

14 Suppose X 1, X 2,..., X n is a random sample from a uniform distribution on (0, θ). Let Y = X (n) and W = X (1). Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

15 Suppose X 1, X 2,..., X n is a random sample from a uniform distribution on (0, θ). Let Y = X (n) and W = X (1). Recall: the uniform density on (0, θ) is 1, if 0 < x < θ, f (x) = θ 0, otherwise, and the distribution function is 0, if x < 0, x F (x) =, if 0 x < θ, θ 1, if x θ. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

16 (cont d) Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

17 (cont d) Hence the probability density function for Y is n ( y ) n 1, if 0 < y < θ, g(y) = θ θ 0, otherwise, { n = θ n y n 1, if 0 < y < θ, 0, otherwise. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

18 (cont d) Hence the probability density function for Y is n ( y ) n 1, if 0 < y < θ, g(y) = θ θ 0, otherwise, { n = θ n y n 1, if 0 < y < θ, 0, otherwise. And the probability density function for W is n ( 1 w ) n 1, if 0 < w < θ, h(w) = θ θ 0, otherwise. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

19 Sample means Theorem If X 1, X 2,..., X n is a random sample from a distribution with mean µ and standard deviation σ, then E[ X ] = µ and var[ X ] = σ2 n. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

20 Sample means Theorem If X 1, X 2,..., X n is a random sample from a distribution with mean µ and standard deviation σ, then E[ X ] = µ and var[ X ] = σ2 n. Proof. We have E[ X ] = E [ 1 n ] n X i = 1 n i=1 n E[X i ] = 1 n (nµ) = µ i=1 and var[ X ] = var [ 1 n ] n X i = 1 n n 2 var[x i ] = 1 n 2 (nσ2 ) = σ2 n. i=1 Dan Sloughter (Furman University) Sampling Distributions March 16, / 10 i=1

21 Sample means (cont d) If we let µ Y denote the mean of a random variable Y and σ 2 Y denote the variance of Y, then the above theorem says µ X = µ and or, equivalently, σ2 σ = 2 X n, σ X = σ n. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

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23 Suppose X 1, X 2,..., X n is a random sample from a Bernoulli distribution with probability of success p. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

24 Suppose X 1, X 2,..., X n is a random sample from a Bernoulli distribution with probability of success p. Let ˆp = X, the sample proportion of successes. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

25 Suppose X 1, X 2,..., X n is a random sample from a Bernoulli distribution with probability of success p. Let ˆp = X, the sample proportion of successes. Then and µˆp = E[X 1 ] = p σˆp = σ n X 1 = p(1 p). n Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

26 Suppose X 1, X 2,..., X n is a random sample from a Bernoulli distribution with probability of success p. Let ˆp = X, the sample proportion of successes. Then and µˆp = E[X 1 ] = p σˆp = σ n X 1 = p(1 p). n Note: for any 0 p 1, 0 p(1 p) 1 4. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

27 Suppose X 1, X 2,..., X n is a random sample from a Bernoulli distribution with probability of success p. Let ˆp = X, the sample proportion of successes. Then and µˆp = E[X 1 ] = p σˆp = σ n X 1 = p(1 p). n Note: for any 0 p 1, 0 p(1 p) 1 4. Hence, for any value of p, σˆp 1 2 n. Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

28 (cont d) Dan Sloughter (Furman University) Sampling Distributions March 16, / 10

29 (cont d) For example, for a sample of size n = 2000, σˆp Dan Sloughter (Furman University) Sampling Distributions March 16, / 10