RANDOM VARIABLE AND JOINT PROBABILITY DISTRIBUTION

RANDOM VARIABLE

• It is an item used to define or denote the outcomes in the sample space known as the sample points.

• It assigns a numerical value to each outcome in the sample space. • It is an item whose numerical value is of a random nature, and

therefore cannot be known with certainty. Types of random variable • Discrete - this type of random variable can only assume a finite or

countable infinite number of possible values • Continuous - this type of random variable can take on any value

within a given range Ex. temperature, volume, weight, diameter, time, quiz average of a

student

PROBABILITY DISTRIBUTION

• a table or a function which helps us determine or compute the probability associated to each value of the random variable

Types of Probability Distribution 1. Discrete probability distribution - one that involves a

discrete random variable

2. Continuous Probability Distribution - one that involves a continuous random variable

DISCRETE PDF Characteristics of a discrete probability distribution 1. f(x) 0 x (for all x)

2. f(x) = 1

3. P(X = x) = f(x) refers to the value of the function when

the random variable X is equal to a specific value x Cumulative Distribution Function • a table or a function that determines the probability that

the random variable X takes on values that are less than or equal to a specific value x

• denoted by: F(x) = P(X x) = f(x)

CONTINUOUS PDF Characteristics of a continuous probability distribution 1. f(x) 0

2. If P(X = c) = 0 then P(a x b) = P(a x b) = P(a x b) = P(a x b)

3. .

4. .

Note: The total area under the curve represents the probability of the entire sample space.

Cumulative Density Function P(x A) = F(A); P(x A) = 1 - F(A); P(A x B) = F(B) - F(A)

b

adxxfbxaP )()(

1)( dxxf

x

dttfxXPxF )(()(

EXPECTATION

EXPECTATION, E(x) or x

• The expected value of the random variable x is the average of all possible values of x or the mean of the x values. It is one of the properties of a probability distribution.

For discrete probability distributions : E(x) = x f(x)

For continuous probability distributions: E(x) = x f(x) dx

VARIANCE VARIANCE, (2

x)

• measures the dispersion or “spread” of the values of x

• the average of the squares of the deviations of all the x values from the mean

• just like x, 2

x is a property of the probability distribution of x

Basic Formula: 2x = E(x - x)

2

Working Formula: 2x = E(x2) - (x)

2 = E(x2) - E(x)2

• Note: x and 2x are measures that provide description to a

population.

Standard Deviation (x): x = (2x)

1/2

Example # 1

Let d represent the number of defectives produced in an hour’s run by a particular automatic machine. The probability distribution of d is given by: 1. What is the probability that in an hour’s run, the machine

will produce at least 3 defectives? 2. Answer (a) if it is known that the machine does produce at

least one defect. 3. Set up F(d) and use it to evaluate P(1 d 4). 4. Find the mean and variance of the number of defects

produced in an hour’s run.

0

)6(

10.0

)(dk

kdxf

elsewhere

d

d

d

5,4

3,2,1

0

Example #2

The random variable x has a density function given by:

1. Find P(0.5 x 2)

2. Find P(x 1.5)

3. Find F(x)

4. Use F(x) to evaluate P(x 2) and P(1 x 2.5)

5. Find and 2

0

)1()(

xkxf

elsewhere

x 20

Example #3

From a box containing 5 red chips and 8 blue chips, three chips are drawn in succession. Find the probability distribution for the number of blue chips selected if:

1. sampling is done with replacement

2. sampling is done without replacement

Example #4 The sales X of a gasoline distributor has a uniform distribution shown in the figure below. Because of daily equipment limitations, sales will never be less than 5,000 gallons per day and never greater than 25,000 gallon per day. Find the probability that the distributor sells 1. at least 20,000 gallons 2. between 15,000 and 23,000 gallons using F(x)

5,000 25,000

f(x)

Example #5

Let X be a random variable whose probability distribution is given by

for 0 X 12

1. Confirm that the function is a probability function

2. Find its expected value and variance and standard deviation

726

1)(

xxf

Assignment The probability distribution of sales of a new drug in units per day is given by:

1. What is the probability of selling 2 units of the drug in

one day? 2. What is the probability that at least 3 units are sold in

one day? 3. What is the probability that 3 units of the drug will be

sold within a period of 2 days? 4. Find F(x) in table form. 5. Use F(x) to evaluate the P(2 x 5) and P(x 3).

0

)( 2x

k

xf

elsewhere

x 6,5,4,3,2,1

Assignment

From a box containing 5 red chips and 8 blue chips, three chips are drawn in succession. Derive the probability distribution of x (where: x = number of trials performed before a blue chip is obtained) if the experiment called for drawing a chip from the box until a blue chip is obtained.

1. assume sampling with replacement

2. assume sampling without replacement

Assignment

A box of a dozen eggs contains 7 good eggs and 5 bad eggs. Mr. Thomas Cook is preparing breakfast for his family - one wife and two kids. He plans to cook an egg for each one of them plus some bacon. He randomly selects 4 eggs from the box and sets these aside in a bowl. 1. Determine the probability distribution function of the

number of good eggs contained in the bowl. 2. What is the probability that the bowl contains at most

one bad egg? 3. What is the probability that Mr. Cook will have to get

eggs from the box again?

Assignment

An operations research analyst has found that the cumulative distribution of a random variable is given by:

F(x) = x2/16 x = 1, 2, 3, 4

Req’d.: Find f(x)

Assignment

Find the cumulative density function of

f(x) = 3X2/125 0 X 5

Evaluate the ff:

1. P(X4)

2. P(X3)

3. P( 2X3.5)

JOINT PROBABILITY DISTRIBUTION

Examples of Joint random variables: • The X number of hours watching TV as related to the Y

number of hours spent exercising. • The X number of books borrowed by a student per

term from the Library and the number of Y Journal articles she photocopies.

• X number of pages on a report and Y grade given by teacher.

• X minutes spent on a sports activity and Y number of blisters created.

• X hours spent at studying and Y grade of student in that course

JOINT PROBABILITY DISTRIBUTION

Let x and y be two different discrete random variables.

f(x, y)

• joint probability distribution functionof 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

DISCRETE JPDF

Characteristics of a Joint Probability Distribution for DISCRETE RANDOM VARIABLE

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

add up to 1. 3. f(x, y) = P(X = x, Y = y) 4. For any region A in the x y plane, P [(x, y) A] = f(x, y)

x

y

MARGINAL DISTRIBUTION

DISCRETE CASE:

g(x) =

h(y) =

y

yxf ),(

x

yxf ),(

CONDITIONAL DISTRIBUTION

DISCRETE CASE:

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

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

)(

),(

)(

),(

xg

yxf

xXP

yYxXP

)(

),(

)(

),(

yh

yxf

yYP

yYxXP

COVARIANCE

Covariance Cov(x,y): A measure of dispersion between joint variables X and Y.

COV(x,y) = E(xy) – E(x) E(y) where:

E(xy) =

E(x) =

E(y) =

y x

yxfyx ),(

)(xxg

)(yyh

Example #6 The following joint probability table for X and Y is presented:

Find 1. g(X) 2. P(X 2|Y=0) 3. h(Y) 4. P(Y 2|X=2) 5. Cov(XY)

Y X

0

2

4

6

0 0.15 0.08 0.05 0.02

1 0.10 0.10 0.15 0.05

2 0.05 0.08 0.05 0.03

3 0.02 0.03 0.02 0.02

Example # 7

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 the joint probability distribution function f(x, y)

1. What is the probability that 1 blue refill will be chosen?

2. What is the probability that a red and a blue will be chosen?

3. What is the probability that no red refill will be selected?

4. What is the probability that I will get at least one blue refill when you did not get any red?

5. Find the Covariance

Assignment

The following joint probability table for X and Y. Find the covariance of X and Y.

Y X

0 1 2 3

0 0.15 0.05 0 0

1 0.05 0.20 0.05 0

2 0 0 0.30 0.20

Assignment

The numbers 1 to 5 are written on pieces of paper and placed inside a drawing box. 2 numbers are picked and their numbers are recorded. Let X be the number of odd numbered papers picked, and Y= the sum of the 2 numbers. Produce the joint probability table.

CONTINUOUS JPDF

Characteristics of a Joint Probability Distribution for CONTINUOUS RANDOM VARIABLE

1. f(x, y) 0

2. f(x, y) dx dy = 1 where U=upper limit; L=lower limit 3. P [ (X, Y) A] = f(x, y) dx dy for 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

U

L U

L

A

MARGINAL DISTRIBUTION

CONTINUOUS CASE:

g(x) =

h(y) =

Uy

Lyf(x, y) dy

Ux

Lxf(x, y) dx

CONDITIONAL DISTRIBUTION

CONTINUOUS CASE:

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

a f (x / y) dx

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

a

f (y / x) dy

Example #8 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) = (2/5)(2x + 3y) 0 x 1, 0 y 1

= 0 elsewhere

1. Find P [ (X, Y) A] where A is the region { (x, y) 0 < x < ½ , ¼0 < y < ½ }

2. Derive g(x) and h(y)

3. Find the conditional probability distribution of X, given that Y = 1 for Problem 1 and use it to evaluate P (0 < x < 0.15 / 0.25<y< 1).

Assignment Jollibee fast food branch operates both a drive-up facility and a walk-in counter. On a randomly selected day, let X = be the proportion of time that the drive up facility is in use (at least one customer is being served or waiting to be served) and Y = be the proportion of time that the walk-in counter is in use. Suppose that the joint pdf of X and Y is given by

f(x,y) = 6/5 (x+y2) for 0 X 1 and 0 Y 1

1. Prove that the function is a legitimate pdf. 2. Give the marginal probability function of X and marginal

probability function of Y 3. What is the probability that both facilities are busy up than to

25% of the time.

Assignment Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose that the actual air pressure in each tire is a random variable X for the right tire, and Y for the left tire, with joint pdf:

f(x,y) = (3/380,000) (X2+Y2) for 20X30 and 20Y30 1. Give the marginal probability distribution of the right tire

alone. 2. Give the probability that the left tire is underfillled. 3. Give the probability that the right tire is underfilled. 4. What is the probability that both tires are underfilled? 5. Are X and Y independent random variables?