Sampling Distributions

1
Sampling Distributions

• A statistic is random in value it changes from
  sample to sample.
• The probability distribution of a statistic is
  called a sampling distribution.
• The sampling distribution can be very useful for
  evaluating the reliability of inference based on
  the statistic.

2
Central Limit Theorem (CLT)

• If a random sample of sufficient size (n30) is
  taken from a population with mean m and variance
  s2gt0, then
• the sample mean will follow a normal distribution
  with mean m and variance s2/n,

3
CLT (continued)

• the sum of the data will follow a normal
  distribution with mean nm and variance ns2.
• The CLT can be used with any sample size if the
  underlying data follows a normal distribution.

4
Standardizing for the CLT

• Z formulae for the CLT include

5
Joint Probability Distributions

• Many situations involve more than one random
  variable of interest. In these cases, a
  multivariate distribution can be used to describe
  the probability distributions of all random
  variables simultaneously.

6
Joint Probability Distributions for Discrete
Random Variables

• Joint Probability Mass Function (pmf)
• Let X and Y be discrete random variables defined
  on the sample space S. The joint pmf, p( x, y )
  is defined for each pair of ( x, y ) to be
• p( x, y ) P( Xx, Yy )
• Note This is equivalent to P( Xx and Yy )

7
Joint PMF

• Properties of a Joint PMF
• 0 p( x, y ) 1
• ?x ?y p( x, y ) 1
• P( ( x, y ) ? A ) ??( x, y ) ? A p( x, y )

8
Marginal PMF

• Let X and Y be discrete random variables with pmf
  p( x, y ) defined on the sample space S. The
  marginal pmf of X is
• pX(x) ?y p( x, y )
• Similarly, the marginal pmf of Y is
• pY(y) ?x p( x, y )

9
Independent Random Variables

• Let X and Y be discrete random variables with pmf
  p( x, y ).
• X and Y are independent if and only if
• p( x, y ) px( x ) py( y ).

10
Extension of Joint PMF to Casewith n Discrete
Random Variables

• For n discrete random variables x1, x2,, xn
• the joint pmf is
• p( x1, x2,, xn ) P( X1x1, X2x2, , Xnxn )
• The marginal distribution for xk is
• p( x1, x2,, xn )
• ?x1 ?x2?xk-1?xk1?xn p( x1, x2,, xn )

11
Joint Probability Distributions for Continuous
Random Variables

• Joint Probability Distribution Function (pdf)
• Let X and Y be continuous random variables
  defined on the sample space S. The joint pdf, f(
  x, y ) is a function such that
• f( x, y ) 0
• ? ? f( x, y ) dy dx 1
• P( ( x, y ) ? A ) ? ? ( x, y ) ? A f( x, y ) dy
  dx

12
Marginal PMF

• Let X and Y be continuous random variables with
  pdf f( x, y ). The marginal pdfs of X and Y are
  
• fX(x) ?y f( x, y ) dy
• Similarly, the marginal pmf of Y is
• fy(y) ?x f( x, y ) dx

13
Independent Random Variables

• Let X and Y be continuous random variables with
  pdf f( x, y ).
• X and Y are independent if and only if
• f( x, y ) fx( x ) fy( y ).

14
Extension of Continuous PDF to Casewith n
Continuous Random Variables

• For n continuous random variables x1, x2, , xn
• the joint pdf f( x1, x2,, xn ) has all the
  properties of a pdf
• The marginal distribution for xk is
• f( x1, x2,, xn )
• ?x1 ?x2 ?xk-1 ?xk1 ?xn f( x1, x2,, xn )

15
Independent Random Variables

• Let X and Y be continuous random variables with
  pdf f( x, y ).
• X and Y are independent if and only if
• f( x, y ) fx( x ) fy( y ).
• X1, X2,, Xn are mutually independent if and only
  if
• f( x1, x2,, xn ) f1(x1) f2(x2) fn (xn)