Joint distributions

1
Joint distributions

• Often times, we are interested in more than one
  random variable at a time.
• For example, what is the probability that a car
  will have at least one engine problem and at
  least one blowout during the same week?
• X of engine problems in a week
• Y of blowouts in a week
• P(X 1, Y 1) is what we are looking for
• To understand these sorts of probabilities, we
  need to develop joint distributions.

2
Discrete distributions

• A discrete joint probability mass function is
  given by
• f(x,y) P(X x, Y y)
• where

3
Return to the car example

• Consider the following joint pmf for X and Y
• P(X 1, Y 1)
• P(X 1)
• E(X Y)

4
Joint to marginals

• The probability mass functions for X and Y
  individually (called marginals) are given by
• Returning to the car example
• fX(x)
• fY(y)
• E(X)
• E(Y)

5
Continuous distributions

• A joint probability density function for two
  continuous random variables, (X,Y), has the
  following four properties

6
Continuous example

• Consider the following joint pdf
• Show condition 2 holds on your own.
• Show P(0 lt X lt 1, ¼ lt Y lt ½) 23/512

7
Joint to marginals

• The marginal pdfs for X and Y can be found by
• For the previous example, find fX(x) and fY(y).

8
Independence of X and Y

• The random variables X and Y are independent if
  f(x,y) fX(x) fY(y) for all pairs (x,y).
• For the discrete clunker car example, are X and Y
  independent?
• For the continuous example, are X and Y
  independent?

9
Sampling distributions

• We assume that each data value we collect
  represents a random selection from a common
  population distribution.
• The collection of these independent random
  variables is called a random sample from the
  distribution.
• A statistic is a function of these random
  variables that is used to estimate some
  characteristic of the population distribution.
• The distribution of a statistic is called a
  sampling distribution.
• The sampling distribution is a key component to
  making inferences about the population.

10
StatCrunch example

• StatCrunch subscriptions are sold for 6 months
  (5) or 12 months (8).
• From past data, I can tell you that roughly 80
  of subscriptions are 5 and 20 are 8.
• Let X represent the amount in of a purchase.
• E(X)
• Var(X)

11
StatCrunch example continued

• Now consider the amounts of a random sample of
  two purchases, X1, X2.
• A natural statistic of interest is X1 X2, the
  total amount of the purchases.

12
StatCrunch example continued

• E(X1 X2)
• E(X1 X22)
• Var(X1 X2)

13
StatCrunch example continued

• If I have n purchases in a day, what is
• my expected earnings?
• the variance of my earnings?
• the shape of my earnings distribution for large
  n?
• Lets experiment by simulating 1000 days with 100
  purchases per day.
• StatCrunch

14
Central Limit Theorem

• We have just illustrated one of the most
  important theorems in statistics.
• As the sample size, n, becomes large the
  distribution of the sum of a random sample from a
  distribution with mean m and variance s2
  converges to a Normal distribution with mean nm
  and variance ns2.
• A sample size of at least 30 is typically
  required to use the CLT
• The amazing part of this theorem is that it is
  true regardless of the form of the underlying
  distribution.

15
Airplane example

• Suppose the weight of an airline passenger has a
  mean of 150 lbs. and a standard deviation of 25
  lbs. What is the probability the combined weight
  of 100 passengers will exceed the maximum
  allowable weight of 15,500 lbs?
• How many passengers should be allowed on the
  plane if we want this probability to be at most
  0.01?

16
The sample mean

• For constant c, E(cY) cE(Y) and Var(cY)
  c2Var(Y)
• E( )
• Var( )
• The CLT says that for large samples, is
  approximately normal with a mean of m and a
  variance of s2/n.
• So, the variance of the sample mean decreases
  with n.

17
Sampling applet