1. Joint Probability Distributions
Outlines
Two Discrete/Continuous Random Variables
Joint Probability Distributions
Marginal Probability Distributions
Conditional Probability Distributions
Independence
Multiple Discrete/Continuous Random Variables
Multinomial Probability Distribution
Covariance and Correlation
Bivariate Normal Distribution
Linear Combination of random variables

2. Joint Probability Distributions
In general, if X and Y are two random variables, the probability distribution that defines their simultaneous behavior is called a joint probability distribution.
For example: X : the length of one dimension of an injection- molded part, and Y : the length of another dimension. We might be interested in
P(2.95  X  3.05 and 7.60  Y  7.80).

3. Two Discrete Random Variables
Joint Probability Distributions
Marginal Probability Distributions
Conditional Probability Distributions
Independence

4. Joint Probability Distributions
The joint probability distribution of two random variables =bivariate probability distribution.
The joint probability distribution of two discrete random variables is usually written as P(X=x, Y=y).

5. Marginal Probability Distributions
Marginal Probability Distribution: the individual probability distribution of a random variable.

6. Marginal Probability Distributions
Example: The marginal probability distribution for X and Y.
y=number of times city name is stated
x=number of bars of signal strength 1 2 3
Marginal probability distribution of Y 4
0.15
0.1
0.05
0.3
0.02
0.17
0.03
0.2
0.25
0.01
0.28
0.55
Marginal probability distribution of X
P(X=3)

7. Conditional Probability Distributions
When two random variables are defined in a random experiment, knowledge of one can change the probabilities of the other.

8. Conditional Mean and Variance

9. Conditional Mean and Variance
Example: From the previous example, calculate
P(Y=1|X=3), E(Y|1), and V(Y|1).

10. Independence
In some random experiments, knowledge of the values of X does not change any of the probabilities associated with the values for Y.
If two random variables are independent, then

11. Multiple Discrete Random Variables
Joint Probability Distributions
Multinomial Probability Distribution

12. Joint Probability Distributions
In some cases, more than two random variables are defined in a random experiment.
Marginal probability mass function

13. Joint Probability Distributions
Mean and Variance

14. Joint Probability Distributions
Conditional Probability Distributions
Independence

15. Multinomial Probability Distribution
A joint probability distribution for multiple discrete random variables that is quite useful in an extension of the binomial.

16. Multinomial Probability Distribution
Example: Of the 20 bits received, what is the probability that 14 are Excellent, 3 are Good, 2 are Fair, and 1 is Poor? Assume that the classifications of individual bits are independent events and that the probabilities of E, G, F, and P are 0.6, 0.3, 0.08, and 0.02, respectively.
One sequence of 20 bits that produces the specified numbers of bits in each class can be represented as: EEEEEEEEEEEEEEGGGFFP
P(EEEEEEEEEEEEEEGGGFFP)=
The number of sequences (Permutation of similar objects)=

17. Two Continuous Random Variables
Joint Probability Distributions
Marginal Probability Distributions
Conditional Probability Distributions
Independence

18. Joint Probability Distributions

19. Joint Probability Distributions
Example: X: the time until a computer server connects to your machine , Y: the time until the server authorizes you as a valid user. Each of these random variables measures the wait from a common starting time and X <Y. Assume that the joint probability density function for X and Y is
The probability that X<1000 and Y<2000 is:

20. Marginal Probability Distributions
Similar to joint discrete random variables, we can find the marginal probability distributions of X and Y from the joint probability distribution.

21. Marginal Probability Distributions
Example: For the random variables in the previous example, calculate the probability that Y exceeds 2000 milliseconds.

22. Conditional Probability Distributions

23. Conditional Probability Distributions
Example: For the random variables in the previous example, determine the conditional probability density function for Y given that X=x
Determine P(Y>2000|x=1500)

24. Conditional Probability Distributions
Mean and Variance

25. Conditional Probability Distributions
Example: For the random variables in the previous example, determine the conditional mean for Y given that x=1500

26. Independence

27. Independence
Example: Let the random variables X and Y denote the lengths of two dimensions of a machined part, respectively.
Assume that X and Y are independent random variables, and the distribution of X is normal with mean 10.5 mm and variance (mm)2 and that the distribution of Y is normal with mean 3.2 mm and variance (mm)2.
Determine the probability that 10.4 < X < 10.6 and 3.15 < Y < 3.25.
Because X,Y are independent

28. Multiple Continuous Random Variables

29. Multiple Continuous Random Variables
Marginal Probability

30. Multiple Continuous Random Variables
Mean and Variance
Independence

31. Covariance and Correlation
When two or more random variables are defined on a probability space, it is useful to describe how they vary together.
It is useful to measure the relationship between the variables.

32. Covariance
Covariance is a measure of linear relationship between the random variables. \
The expected value of a function of two random variables
h(X, Y ).

33. Covariance

34. Covariance

35. Covariance
Example: For the discrete random variables X, Y with the joint distribution shown in Fig. Determine

36. Correlation
The correlation is a measure of the linear relationship between random variables.
Easier to interpret than the covariance.

37. Correlation
For independent random variables

38. Correlation
Example: Two random variables , calculate the covariance and correlation between X and Y.

39. Bivariate Normal Distribution
Correlation

40. Bivariate Normal Distribution
Marginal distributions
Dependence

41. Bivariate Normal Distribution
Conditional probability

42. Bivariate Normal Distribution
Ex. Suppose that the X and Y dimensions of an injection-modeled part have a bivariate normal distribution with Find the P(2.95<X<3.05,7.60<Y<7.80)

43. Bivariate Normal Distribution
Ex. Let X, Y : milliliters of acid and base needed for equivalence, respectively. Assume X and Y have a bivariate normal distribution with
Covariance between X and Y
Marginal probability distribution of X
P(X<116)
P(X|Y=102)
P(X<116|Y=102)

44. Linear Combination of random variables

45. Linear Combination of random variables
Mean and Variance

46. Linear Combination of random variables
Ex. A semiconductor product consists of 3 layers. The variances in thickness of the first, second, and third layers are 25,40,30 nm2 . What is the variance of the thickness of the final product? Let X1, X2, X3, and X be random variables that denote the thickness of the respective layers, and the final product. V(X)=V(X1)+V(X2)+V(X3)= =95 nm2

47. Homework
The time between surface finish problems in a galvanizing process is exponentially distributed with a mean of 40 hours. A single plant operates three galvanizing lines that are assumed to operate independently.
What is the probability that none of the lines experience a surface finish problem in 40 hours of operation?
What is the probability that all three lines experience two surface finish problems between 20 and 40 hours after starting the operation?
Suppose X and Y have a bivariate normal distribution with
Determine the following:
a) P(2.95<X<3.05)
b) P(7.60<Y<7.80)
c) P(2.95<X<3.05,7.60<Y<7.80)