Lecture 4. August 24, Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University.

Transcription

1 random Lecture 4 Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University August 24, 2007

2 random random

3 random 1 Define random 2 and 3 4 Co and 5 Standard error of the mean 6 Unbiasedness of the sample

4 random are simply random collected into a vector For example if X and Y are random (X, Y ) is a random vector Joint density f (x, y) satisfies f > 0 and f (x, y)dxdy = 1 For discrete random f (x, y) = 1 In this lecture we focus on independent random where f (x, y) = f (x)g(y)

5 random Two A and B are independent if P(A B) = P(A)P(B) Two random, X and Y are independent if for any two sets A and B P([X A] [Y B]) = P(X A)P(Y B) If A is independent of B then A c is independent of B A is independent of B c A c is independent of B c

6 Example random What is the probability of getting two consecutive heads? A = {Head on flip 1} P(A) =.5 B = {Head on flip 2} P(B) =.5 A B = {Head on flips 1 and 2} P(A B) = P(A)P(B) =.5.5 =.25

7 Example random Volume 309 of Science reports on a physician who was on trial for expert testimony in a criminal trial Based on an estimated prevalence of sudden infant death syndrome of 1 out of 8, 543, Dr Meadow testified that that the probability of a mother having two children with SIDS was ( 1 8,8543 ) 2 The mother on trial was convicted of murder What was Dr Meadow s mistake(s)?

8 Example: continued random For the purposes of this class, the principal mistake was to assume that the probabilities of having SIDs within a family are independent That is, P(A 1 A 2 ) is not necessarily equal to P(A 1 )P(A 2 ) Biological processes that have a believed genetic or familiar environmental component, of course, tend to be dependent within families In addition, the estimated prevalence was obtained from an unpublished report on single cases; hence having no information about recurrence of SIDs within families

9 Useful fact random We will use the following fact extensively in this class: If a collection of random X 1, X 2,..., X n are independent, then their joint distribution is the product of their individual densities or mass functions That is, if f i is the density for random variable X i we have that n f (x 1,..., x n ) = f i (x i )

10 random In the instance where f 1 = f 2 =... = f n we say that the X i are iid for independent and identically distributed iid random are the default model for random samples Many of the important theories of statistics are founded on assuming that are iid

11 Example random Suppose that we flip a biased coin with success probability p n times, what is the join density of the collection of outcomes? These random are iid with densities p x i (1 p) 1 x i Therefore f (x 1,..., x n ) = n p x i (1 p) 1 x i = p x i (1 p) n x i

12 random The co between two random X and Y is defined as Cov(X, Y ) = E[(X µ x )(y µ y )] = E[XY ] E[X ]E[Y ] The following are useful facts about co 1 Cov(X, Y ) = Cov(Y, X ) 2 Cov(X, Y ) can be negative or positive 3 Cov(X, Y ) Var(X )Var(y)

13 random The between X and Y is Cor(X, Y ) = Cov(X, Y )/ Var(X )Var(y) 1 1 Cor(X, Y ) 1 2 Cor(X, Y ) = ±1 if and only if X = a + by for some constants a and b 3 Cor(X, Y ) is unitless 4 X and Y are uncorrelated if Cor(X, Y ) = 0 5 X and Y are more positively correlated, the closer Cor(X, Y ) is to 1 6 X and Y are more negatively correlated, the closer Cor(X, Y ) is to 1

14 random useful results Let {X i } n be a collection of random When the {X i } are uncorrelated ( n ) n Var a i X i + b = ai 2 Var(X i ) Otherwise = ( n ) Var a i X i + b n n 1 ai 2 Var(X i ) + 2 n a i a j Cov(X i, X j ) If the X i are iid with σ 2 then Var( X ) = σ 2 /n and E[S 2 ] = σ 2 j=i

15 random Example proof Prove that Var(X + Y ) = Var(X ) + Var(Y ) + 2Cov(X, Y ) Var(X + Y ) = E[(X + Y )(X + Y )] E[X + Y ] 2 = E[X 2 + 2XY + Y 2 ] (µ x + µ y ) 2 = E[X 2 + 2XY + Y 2 ] µ 2 x 2µ x µ y µ 2 y = (E[X 2 ] µ 2 x) + (E[Y 2 ] µ 2 y ) + 2(E[XY ] µ x µ y ) = Var(X ) + Var(Y ) + 2Cov(X, Y )

16 random Result A commonly used subcase from these is that if a collection of random {X i } are uncorrelated, then the of the sum is the sum of the s ( n ) n Var X i = Var(X i ) Therefore, it is sums of s that tend to be useful, not sums of standard deviations; that is, the standard deviation of the sum of bunch of independent random is the square root of the sum of the s, not the sum of the standard deviations

17 random Suppose X i are iid with σ 2 Var( X ) = Var mean ( 1 n ) n X i = 1 n 2 Var ( n X i ) = 1 n n 2 Var(X i ) = 1 n 2 nσ2 = σ2 n

18 comments random When X i are independent with a common Var( X ) = σ2 n σ/ n is called the standard error of the sample mean The standard error of the sample mean is the standard deviation of the distribution of the sample mean σ is the standard deviation of the distribution of a single observation Easy way to remember, the sample mean has to be less variable than a single observation, therefore its standard deviation is divided by a n

19 random is defined as n S 2 = (X i X ) 2 n 1 is an estimator of σ 2 The numerator has a version that s quicker for calculation n (X i X ) 2 = n X 2 i n X 2 is (nearly) the mean of the squared deviations from the mean

20 random [ n ] E (X i X ) 2 is unbiased = = = n E [ Xi 2 ] [ ne X 2] n { Var(Xi ) + µ 2} n { Var( X ) + µ 2} n { σ 2 + µ 2} n { σ 2 /n + µ 2} = nσ 2 + nµ 2 σ 2 nµ 2 = (n 1)σ 2

21 Hoping to avoid some confusion random Suppose X i are iid with mean µ and σ 2 S 2 estimates σ 2 The calculation of S 2 involves dividing by n 1 S/ n estimates σ/ n the standard error of the mean S/ n is called the sample standard error (of the mean)

22 Example random In a study of 495 organo-lead workers, the following summaries were obtained for TBV in cm 3 mean = sum of squared observations = sample sd = ( )/494 = estimated se of the mean = / 495 = 5.062