Download Week 3 - The University of Texas at Dallas

Random Variables (Chapter 2)
Random variable = A real-valued function of an outcome
X = f(outcome)
Domain of X: Sample space of the experiment.
Ex: Consider an experiment consisting of 3 Bernoulli trials.
Bernoulli trial = Only two possible outcomes – success (S) or failure (F).
•
•
•
“IF” statement: if … then “S” else “F”
Examine each component. S = “acceptable”, F = “defective”.
Transmit binary digits through a communication channel. S = “digit received correctly”, F = “digit
received incorrectly”.
Suppose the trials are independent and each trial has a probability ½ of success.
X = # successes observed in the experiment.
Possible values:
Outcome Value of X
(SSS)
(SSF)
(SFS)
…
…
(FFF)
Random variable:
•
•
•
•
•
Assigns a real number to each outcome in S.
Denoted by X, Y, Z, etc., and its values by x, y, z, etc.
Its value depends on chance.
Its value becomes available once the experiment is completed and the outcome is known.
Probabilities of its values are determined by the probabilities of the outcomes in the sample space.
Probability distribution of X = A table, formula or a graph that summarizes how the total probability of
one is distributed over all the possible values of X.
In the Bernoulli trials example, what is the distribution of X?
1
Two types of random variables:
Discrete rv = Takes finite or countable number of values
•
•
•
Number of jobs in a queue
Number of errors
Number of successes, etc.
Continuous rv = Takes all values in an interval – i.e., it has uncountable number of values.
•
•
•
•
Execution time
Waiting time
Miles per gallon
Distance traveled, etc.
Discrete random variables
X = A discrete rv.
•
•
•
•
Probability mass function (PMF) of X = Probability distribution of X.
Notation: p(x) = P(X = x) = probability that the rv X takes the value x.
Once we have the PMF, we can compute any probability of interest.
0 ≤ p(x) ≤ 1, total probability = ∑ x p ( x) = 1
Ex: “Pick Six” in Texas Lottery. In this Lottery, a player picks six numbers from the numbers 1 through
50 with no repetitions and pays $1.00. On Wednesday evenings, the Texas State Lottery Commission
televises one of their employees randomly picking six balls without replacement, each with a number
from 1 to 50 on it, from a large hopper. The player is paid if his/her number matches with the selected
balls for three or more numbers. Suppose X = # of matches a player has with the selected balls.
(a) What is the PMF of X?
Value of X
# of outcomes in sample space
(# of matches)
that give this value of X
0
1
2
3
4
5
6
Total
p(x)
=
0.44422536452
=
0.41005418264
=
0.12814193207
=
0.01666886921
=
0.00089297514
=
0.00001661349
=
0.00000006293
1.00000000000
2
In this example, in general,
p(x) = P(X = x) =
(b) What is the probability of winning nothing?
Cumulative distribution function (CDF) of X is defined as
F ( x) = P( X ≤ x) = ∑ y ≤ x p( y )
•
•
•
•
•
Gives the total probability on the left of x or the probability that the observed value of X is at most x.
F(x) is non-decreasing.
Jumps by p(x) at the point x.
F(– ∞) = 0, and F(+ ∞) = 1
P( a < X ≤ b) = F(b) – F(a)
Ex: Consider a rv X whose CDF is the following.
x<1
 0
1/ 6 1 ≤ x < 2
2 / 6 2 ≤ x < 3

F(x) = 3/ 6 3 ≤ x < 4
4 / 6 4 ≤ x < 5
5/ 6 5 ≤ x < 6
 1
6≤x

(a) What is the PMF of X?
(b) Find P(X will be greater than 3, but no more than 5).
3
Random vectors and joint distributions
If X, Y = random variables then (X, Y) = random vector.
It has joint PMF p(x, y) = P{(X, Y) = (x, y)} = P(X = x and Y = y)
•
•
•
0 ≤ p(x, y) ≤ 1
p ( x, y ) = total probability = 1
P[(X, Y) ∈ A] = ∑( x , y )∈A p ( x, y )
∑( x , y )
From the joint PMF, we can obtain the marginal PMF’s of X and Y:
From the marginal PMF’s, we may not be able to obtain the joint PMF.
Ex: Consider a program with two modules, having module execution times X and Y minutes,
respectively. The following table describes their joint PMF:
p(x,y)
y=1 y=2 y=3 y=4
x=1 1/4 1/16 1/16 1/8
x=2 1/16 1/8 1/4 1/16
(a) Find the probability that P(Y ≥ X).
(b) Suppose that the two modules are run concurrently. Find the probability that the total execution time
of the program is 3 minutes.
4
(c) Find the marginal PMF of X.
Independent random variables
Random variables X and Y are independent if for every pair (x, y)
p ( x, y ) =
i.e., {X = x} and {Y = y} are independent events for all (x, y).
•
•
•
Two rv’s are independent if their joint PMF is the product of their marginals.
To show independence, verify the above equality for all (x, y).
To show dependence, find one pair (x, y) that violates it.
Ex cont’d: The marginal PMF’s of X and Y are:
1
2
x
P(X=x) 0.5 0.5
1
2
3
4
y
P(Y=y) 5/16 3/16 5/16 3/16
Are X and Y independent?
Note: The concept of joint distribution and independence can be extended to the case of more
than two rv’s. See sections 2.8-2.9 of the text.
5