1. Chapter 5 Joint Probability Distributions and Random Samples
5.1 - Jointly Distributed Random Variables
5.2 - Expected Values, Covariance, and Correlation
5.3 - Statistics and Their Distributions
5.4 - The Distribution of the Sample Mean
5.5 - The Distribution of a Linear Combination

2. Population Distribution of X
Suppose X ~ N(μ, σ), then…
X = Age of women in U.S. at first birth X
Density
Each of these individual ages x is a particular value of the random variable X. Most are in the neighborhood of μ, but there are occasional outliers in the tails of the distribution.     x4 x5 x1 x2  x3
… etc….
σ = 1.5 x x x x x
μ = 25.4

3. Population Distribution of X
Suppose X ~ N(μ, σ), then…
Sample,
n = 400
Sample,
n = 400
X = Age of women in U.S. at first birth
Sample,
n = 400
Sample,
n = 400 X
Density
Sample,
n = 400
How are these values distributed?
… etc….
Each of these sample mean values is a “point estimate” of the population mean μ…
σ = 1.5
μ = 25.4

4. Population Distribution of X
Sampling Distribution of
for any sample size n.
Suppose X ~ N(μ, σ), then…
Suppose X ~ N(μ, σ), then…
X = Age of women in U.S. at first birth
Density
μ = X
Density
μ =
σ = 1.5
“standard error”
The vast majority of sample means are extremely close to μ, i.e., extremely small variability.
How are these values distributed?
… etc….
Each of these sample mean values is a “point estimate” of the population mean μ…
μ = 25.4

5. Population Distribution of X
Sampling Distribution of
Suppose X ~ N(μ, σ), then…
Suppose X ~ N(μ, σ), then…  
for large sample size n.
for any sample size n.
X = Age of women in U.S. at first birth
Density
μ = X
Density
μ =
σ = 2.4
Suppose X ~ N(μ, σ), then…
“standard error”
The vast majority of sample means are extremely close to μ, i.e., extremely small variability.
… etc….
Each of these sample mean values is a “point estimate” of the population mean μ…
μ = 25.4

6. Population Distribution of X
Sampling Distribution of
X ~ Anything with finite μ and σ
Suppose X  N(μ, σ), then… 
for large sample size n.
for any sample size n.
X = Age of women in U.S. at first birth
Density
μ = X
Density
μ =
σ = 2.4
Suppose X ~ N(μ, σ), then…
“standard error”
The vast majority of sample means are extremely close to μ, i.e., extremely small variability.
… etc….
Each of these sample mean values is a “point estimate” of the population mean μ…
μ = 25.4

7. Density  
Density 
“standard error”

8. Probability that a single house selected at random costs less than $300K = ?
= Cumulative area under density curve for X up to 300.
= Z-score
Density  
Density 
“standard error”
Example:
X = Cost of new house ($K) 
300

9. Probability that a single house selected at random costs less than $300K = ?
0.6554
= Z-score
Density  
Density 
“standard error”
Example:
X = Cost of new house ($K) 
300

10. Probability that a single house selected at random costs less than $300K = ?
0.6554
= Z-score
Probability that the sample mean of n = 36 houses selected at random is less than $300K = ?
= Cumulative area under density curve for up to 300.
Density  
Density 
“standard error”
Example:
X = Cost of new house ($K)
$12.5K 
300
300

11. Probability that a single house selected at random costs less than $300K = ?
0.6554
= Z-score
Probability that the sample mean of n = 36 houses selected at random is less than $300K = ?
0.9918
= Z-score
Density  
Density 
“standard error”
Example:
X = Cost of new house ($K)
$12.5K 
300
300

12.      large Density Density “standard error” mild skew
approximately 
large
Density 
Density
“standard error”
mild skew   

13. ~ CENTRAL LIMIT THEOREM ~
approximately
continuous or discrete, 
as n  ,
large  
Density 
Density
“standard error”  

14. ~ CENTRAL LIMIT THEOREM ~
continuous or discrete,
approximately 
as n  ,
large  
Density 
Example:
X = Cost of new house ($K)
Density
“standard error”  

15. Probability that a single house selected at random costs less than $300K = ?
= Cumulative area under density curve for X up to 300.
Probability that the sample mean of n = 36 houses selected at random is less than $300K = ?
0.9918
= Z-score
Density 
“standard error”
Example:
X = Cost of new house ($K)
Density
$12.5K 
300
300 

17. x
p(x)
0.5 10
0.3 20
0.2

18. .25 5
.30 = 10
.29 = 15
.12 = 20
.04

21. possibly log-normal… More on CLT…
but remember Cauchy and 1/x2, both of which had nonexistent …
CLT may not work!
More on CLT…
heavily skewed tail
each based on 1000 samples

22. Population Distribution of X
Density
Population Distribution of X
X ~ Dist(μ, σ)
Random Variable
More on CLT…
X = Age of women in U.S. at first birth 
If this first individual has been randomly chosen, and the value of X measured, then the result is a fixed number x1, with no random variability… and likewise for x2, x3, etc. DATA!
BUT…

23. Population Distribution of X
Density
Population Distribution of X
X ~ Dist(μ, σ)
Random Variable
More…
X = Age of women in U.S. at first birth 
If this first individual has been randomly chosen, and the value of X measured, then the result is a fixed number x1, with no random variability… and likewise for x2, x3, etc. DATA!
However, if this is not the case, then this first “value” of X is unknown, thus can be considered as a random variable X1 itself… and likewise for X2, X3, etc.
BUT…
The collection {X1, X2, X3, …, Xn} of “independent, identically-distributed” (i.i.d.) random variables is said to be a random sample.

24. Population Distribution of X
Density
Population Distribution of X
X ~ Dist(μ, σ)
Random Variable
Sample,
size n
More…
X = Age of women in U.S. at first birth
CENTRAL LIMIT THEOREM
Sampling Distribution of
etc……
Density
Claim:
for any n
Proof:

25. Population Distribution of X
Density
Population Distribution of X
X ~ Dist(μ, σ)
Random Variable
More…
X = Age of women in U.S. at first birth
CENTRAL LIMIT THEOREM
Sampling Distribution of
etc……
Density
Claim:
for any n
Proof:

26. Recall… More on CLT… Normal Approximation to the Binomial Distribution
continuous
discrete
Recall…
More on CLT…
Normal Approximation to the Binomial Distribution
Suppose a certain outcome exists in a population, with constant probability .
We will randomly select a random sample of n individuals, so that the binary “Success vs. Failure” outcome of any individual is independent of the binary outcome of any other individual, i.e., n Bernoulli trials (e.g., coin tosses).
Then X is said to follow a Binomial distribution, written X ~ Bin(n, ), with “probability function”
p(x) = , x = 0, 1, 2, …, n.
P(Success) = 
P(Failure) = 1 – 
Discrete random variable
X = # Successes (0, 1, 2,…, n) in a random sample of size n

27. Normal Approximation to the Binomial Distribution CLT
continuous
discrete
Normal Approximation to the Binomial Distribution
Suppose a certain outcome exists in a population, with constant probability .
We will randomly select a random sample of n individuals, so that the binary “Success vs. Failure” outcome of any individual is independent of the binary outcome of any other individual, i.e., n Bernoulli trials (e.g., coin tosses).
Then X is said to follow a Binomial distribution, written X ~ Bin(n, ), with “probability function”
p(x) = , x = 0, 1, 2, …, n.
P(Success) = 
P(Failure) = 1 – 
CLT
See Prob 5.3/7
Discrete random variable
X = # Successes (0, 1, 2,…, n) in a random sample of size n

28. Normal Approximation to the Binomial Distribution CLT
continuous
discrete ??
Normal Approximation to the Binomial Distribution
Suppose a certain outcome exists in a population, with constant probability .
We will randomly select a random sample of n individuals, so that the binary “Success vs. Failure” outcome of any individual is independent of the binary outcome of any other individual, i.e., n Bernoulli trials (e.g., coin tosses).
Then X is said to follow a Binomial distribution, written X ~ Bin(n, ), with “probability function”
p(x) = , x = 0, 1, 2, …, n.
P(Success) = 
P(Failure) = 1 – 
CLT
Discrete random variable
X = # Successes (0, 1, 2,…, n) in a random sample of size n