JOINT PROBABILITY DISTRIBUTION

Let x and y be two different discrete random variables.

f(x, y) - joint probability distribution of x and y

-probability distribution of the simultaneous occurrence of x and y; i.e., f(x, y) = P(X = x, Y = y)

-gives the probability distribution that outcomes x and y can occur at the same time

-For example,

Let x - age to the nearest year of a TV set that is to be repaired

y - number of defective tubes in the set

f(x, y) = f(5, 3) = probability that the TV set is 5 years old and needs 3

new tubes

Characteristics of a Joint Probability Distribution

  1. f(x, y)  0 for all (x, y)
  2. f(x, y) = 1add up the probabilities of all possible combinations of x

and y within the range

  1. f(x, y) = P(X = x, Y = y)
  2. For any region A in the x y plane, P [(x, y)  A] =  f(x, y)

Example 1:

Two refills for a ballpoint pen are selected at random from a box containing 3 blue refills, 2 red refills and 3 green refills. If X is the number of blue refills and Y is the number of red refills selected, find

  1. the joint probability distribution function f(x, y)
  2. P [(X, Y)  A] , where A is the region { (x, y)  x + y  1 }

JOINT DENSITY FUNCTION

Joint Density Function – joint distribution of continuous random variables

Characteristics of a Joint Density Function

  1. f(x, y)  0
  1. f(x, y) dx dy = 1
  1. P [ (X, Y)  A] = f(x, y) dx dyfor any region A in the x y plane

Note:f(x, y) - surface lying above the x y plane

Probability - volume of the right cylinder bounded by the base A and the surface

Example 2:

A candy company distributes boxes of chocolates with a mixture of creams, toffees and nuts coated in both light and dark chocolate. For a randomly selected box, let X and Y, respectively be the proportion of the light and dark chocolates that are creams and suppose that the joint density function is given by:

f(x, y)=k(2x + 3y)0  x  1,0  y  1

0elsewhere

Find P [ (X, Y)  A] where A is the region { (x, y) 0 < x < ½ , ¼ < y < ½ }

NOTE:For the discrete case,P(X = x, Y = y) = f(x, y)

ex.P(x = 2, y = 1) = f(2, 1)

For the continuous case, P(X = x, Y = y)  f(x, y)

MARGINAL DISTRIBUTIONS

Given the joint probability distribution f(x, y) of the discrete random variable X and Y, the probability distribution g(x) of X along is obtained by summing f(x, y) over the values of y. Similarly, the probability distribution h(y) of Y alone is obtained by summing f(x, y) over the values of x. g(x) and h(y) are defined to be the marginal distributions of x and y respectively.

g(x) = f(x, y)h(y) = f(x, y)for the discrete case

g(x) = f(x, y) dyh(y) = f(x, y) dx for the continuous case

Example 3:

Derive g(x) and h(y) for Example 1.

Example 4:

Derive g(x) and h(y) for the joint density function in Example 2.

CONDITIONAL DISTRIBUTIONS

Recall:Conditional Probability Formula

P ( B / A) = P(A B)

P(A)

Consider 2 random variables X and Y:

If we let A be the event defined by X = x and B be the event that Y = y, we have,

P ( Y= y) / X = x ) = P (X = x, Y = y)

P (X = x)

= f(x, y)

g(x)g(x) > 0

where X and Y are discrete random variables

P (Y = y / X = x ) may actually be expressed as a probability distribution denoted by f( y / x). Therefore, f (y / x) is called by conditional distribution of the random variable Y given that X = x.

Generalization

Let X and Y be two random variables, discrete or continuous. The conditional probability distribution of the random variable Y given that X = x, is given by

f (y / x) = f(x, y)g(x) > 0

g(x)

(pure function of y)

Similarly, the conditional probability distribution of the random variable X given that Y = y, is given by

f (x / y) = f(x, y)h(y) > 0

h(y)

(pure function of x)

Note:f (x / y) only gives P ( X = x / Y = y). If one wishes to find the probability that the discrete random variable x falls between a and b when it is known that the discrete variable Y = y, then we evaluate

P (a < x < b / Y = y) = f (x / y)

Similarly,

P (a < y < b / X = x) = f (x / y)

For the continuous case:

P (a < x < b / Y = y) = f (x / y) dx

P (a < y < b / X = x) = f (y / x) dy

Example 5:

Find the conditional probability distribution of X, given that Y = 1 for Example 1 and use it to evaluate P (x = 0 / y = 1).

STATISTICAL INDEPENDENCE

Recall:P (B / A) = P(A B)

P(A)

P(A B) = P(A) * P (B / A)

P(A B) = P(A) * P (B) if A and B are statistically independent

Similarly,

f(y / x) = f(x, y)

g(x)

f(x, y) = g(x) * f (y / x)

f(x, y) = g(x) * h(y)if X and Y are statistically independent

OR:f(y / x) = f(x, y)

g(x)

f(x, y) = g(x) * f (y / x)

h(y) = f(x, y) dx= g(x) * f(y / x) dx

pure function of y

if x and y are independent

h(y) = f (y / x) g(x) dx

h(y) = f(x, y) / g(x)

f(x, y) = g(x) * h(y)

Let X and Y be two random variables, discrete or continuous, with joint probability distribution f(x, y) and marginal distributions g(x) and h(y), respectively. The random variable X and Y are said to be statistically independent if and only if

f(x, y) = g(x) * h(y)for all (x, y) within their range

QUAMETH Notes: Joint Probability DistributionPage 1 of 4