Joint Probability Distributions Given two random variables X
and Y that are defined on the same probability space, the Joint
Distribution for X and Y defines the probability of events defined
in terms of both X and Y. In the case of only two random variables,
this is called a Bivariate Distribution. The concept generalizes to
any number of random variables, giving a Multivariate
Distribution.
Slide 3
Example Consider the roll of a die and let A = 1 if the number
is even (2, 4, or 6) and A = 0 otherwise. Furthermore, let B = 1 if
the number is prime (2, 3 or 5) and B = 0 otherwise. Find the Joint
Distribution of A and B? P(A = 0, B = 0) = {1} = 1/6 P(A = 0, B =
1) = {3, 5} = 2/6 P(A = 1, B = 0) = {4, 6} = 2/6 P(A = 1, B = 1) =
{2} = 1/6
Slide 4
Example YXYX Y 1 = 0Y 2 = 1 Rows Total X 1 = 0 1/62/63/6 = 1/2
X 2 = 12/61/63/6 = 1/2 Column Total3/6 = 1/2 1
Slide 5
Joint Probability Distributions
Slide 6
Discrete Joint Probability Distribution X and Y are Independent
f(x, y) = f(x) x f(y) X and Y are Dependent
Slide 7
Discrete Joint Probability Distribution A die is flipped and a
coin is tossed YXYX 123456Row Totals Headf(H, 1) = 1/12 1/2
Tailf(T, 1) = 1/12 1/2 Column Totals 1/6 1
Slide 8
Marginal Probability Distributions Marginal Probability
Distribution: the individual probability distribution of a random
variable.
Slide 9
Discrete Joint Probability Distribution A die is flipped and a
coin is tossed YXYX 123456Marginal Probability of X Headf(H, 1) =
1/12 1/2 Tailf(T, 1) = 1/12 1/2 Marginal Probability of Y 1/6
1
Slide 10
Joint Probability Distributions
Slide 11
Continuous Two Dimensional Distribution
Slide 12
Joint Probability Distributions 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
Problem 3 Let f(x, y) = k. If x > 0, y > 0, x +y < 3
and zero otherwise. Find k. Find P(X + Y 1) and P(Y > X).
Slide 22
Problem 7 What are the mean thickness and standard deviation of
transfer cores each consisting of 50 layers of sheet metal and 49
insulating paper layers, if the metal sheets have mean thickness
0.5 mm each with a standard deviation of 0.05 mm and paper layers
have mean thickness 0.05mm each with a standard deviation of 0.02
mm?
Slide 23
Problem 9 A 5-gear Assembly is put together with spacers
between the gears. The mean thickness of the gears is 5.020 cm with
a standard deviation of 0.003 cm. The mean thickness of spacers is
0.040 cm with a standard deviation of 0.002 cm. Find the mean and
standard deviation of the assembled units consisting of 5 randomly
selected gears and 4 randomly selected spacers.
Slide 24
Problem 11 Show that the random variables with the density f(x,
y) = x + y and g(x, y) = (x+1/2)(y+1/2) If 0 x 1 and 0 y 1 and f(x,
y) = 0 and g(x, y) = 0 and zero otherwise, have the same Marginal
Distribution.
Slide 25
Problem 13 An electronic device consists of two components. Let
X and Y [months] be the length of time until failure of the first
and second components, respectively. Assume that X and Y have the
Probability Density f(x, y) = 0.01e -0.1(x + y) If x > 0 and y
> 0 and zero otherwise. a. Are X and Y dependent or independent?
b. Find densities of Marginal Distribution. c. What is the
probability that the first component has a lifetime of 10 months or
longer?
Slide 26
Problem 15 Find P(X > Y) when (X, Y) has the Probability
Density f(x, y) = 0.25e -0.5(x + y) If x 0, y 0 and zero
otherwise.
Slide 27
Problem 17 Let (X, Y) have the Probability Function f(0, 0) =
f(1, 1) = 1/8 f(0, 1) = f(1, 0) = 3/8 Are X and Y independent?
Slide 28
Marginal Probability Distributions Example: For the random
variables in the previous example, calculate the probability that Y
exceeds 2000 milliseconds.
Slide 29
Conditional Probability Distributions When two random variables
are defined in a random experiment, knowledge of one can change the
probabilities of the other.
Slide 30
Conditional Mean and Variance
Slide 31
Example: From the previous example, calculate P(Y=1|X=3),
E(Y|1), and V(Y|1).
Slide 32
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
Slide 33
Multiple Discrete Random Variables Joint Probability
Distributions Multinomial Probability Distribution
Slide 34
Joint Probability Distributions In some cases, more than two
random variables are defined in a random experiment. Marginal
probability mass function
Slide 35
Joint Probability Distributions Mean and Variance
Slide 36
Joint Probability Distributions Conditional Probability
Distributions Independence
Slide 37
Multinomial Probability Distribution A joint probability
distribution for multiple discrete random variables that is quite
useful in an extension of the binomial.
Slide 38
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)=
Slide 39
Two Continuous Random Variables Joint Probability Distributions
Marginal Probability Distributions Conditional Probability
Distributions Independence
Slide 40
Conditional Probability Distributions
Slide 41
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)
Slide 42
Conditional Probability Distributions Mean and Variance
Slide 43
Conditional Probability Distributions Example: For the random
variables in the previous example, determine the conditional mean
for Y given that x=1500
Slide 44
Independence
Slide 45
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 0.0025 (mm) 2 and that the
distribution of Y is normal with mean 3.2 mm and variance 0.0036
(mm) 2. Determine the probability that 10.4 < X < 10.6 and
3.15 < Y < 3.25. Because X,Y are independent
Slide 46
Multiple Continuous Random Variables
Slide 47
Marginal Probability
Slide 48
Multiple Continuous Random Variables Mean and Variance
Independence
Slide 49
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.
Slide 50
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 ).
Slide 51
Covariance
Slide 52
Slide 53
Example: For the discrete random variables X, Y with the joint
distribution shown in Fig. Determine
Slide 54
Correlation The correlation is a measure of the linear
relationship between random variables. Easier to interpret than the
covariance.
Slide 55
Correlation For independent random variables
Slide 56
Correlation Example: Two random variables, calculate the
covariance and correlation between X and Y.
Slide 57
Bivariate Normal Distribution Correlation
Slide 58
Bivariate Normal Distribution Marginal distributions
Dependence
Slide 59
Bivariate Normal Distribution Conditional probability
Slide 60
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