Random
Variables
Random
Variables
A random variable, usually written X, is a
variable whose possible values are
numerical outcomes of a random
phenomenon
example
Flip a coin three times; X = the total number of heads.The values o
X are .
X= 0, 1, 2, 3
Throw two dice; X = the sum of the numbers facing up.The
values of X are .
X= 2, 3, 4, …, 12
Throw one die over and over until you get a six; X = the number of
X=1, 2, 3, 4, ...
Types of Random
Variables
Discrete random variables are ones that have a finite or
countable number of possible outcomes (like number
of heads when flipping several coins).
Continuous random variables are ones that have an
infinite or uncountable number of possible outcomes
(like your exact speed on the highway, or how far
someone jumps)
Probability with discrete variables
Throw a pair of fair dice, and take X to be the sum of the
numbers facing up. X= 2, 3, 4, … ,12
◦ The event that X = 2 is {(1, 1)}
The event that you throw a 2
◦ The event that X = 3 is {(2, 1), (1, 2)}
The event that you throw a 3
◦ The event that X = 4 is {(3, 1), (2, 2), (1, 3)} The event that you throw a 4
◦ P(X = 4) = 1/12
◦ The probability that X = 4 is
1/12
practice
Throw a pair of fair dice,
and take X to be the sum of
the numbers facing up.
X= 2, 3, 4, … ,12
P(X=1) =
0
P(X=2) = 1/36
P(X=3) = 2/36
P(X=4) = 3/36
Problem 2: Tossing a coin 4 times.
Find the discrete probability of the
following:
let X= number of heads
P(X=0) = 1/16
P(X=1) = 4/16
P(X=2) = 6/16
P(X=3) = 4/16
P(X=4) = 1/16
Problem 2: Tossing a coin 4 times. Find the
probability
of
the
following:
Where: X= number of heads
Number of heads
Probability
0
1
2
3
4
1/16 4/16 6/16 4/16 1/16
What is:
P(X ≤ 2)
=
P(x=0) + P(x=1) + P(x=2)
=
1/16 + 4/16 + 6/16
=
11/16
Or 0.6875
Education
Level:
A study
of education followed a large group of fifthgrade children to see how many years of school they
eventually completed. Let X be the highest year of
school that a randomly chosen 5th graders completes
Years
probability
4
5
6
7
8
9
10
11
12
0.010 0.007 0.007 0.013 0.032 0.068 0.070 0.041 0.752
A. Is this a “legit” continuous probability
distribution?
Yes, because the probabilities add up to 1
B. What is the percent of 5th graders eventually finished 12th grade?
There are 75.2% 5th graders eventually finished 12th grade.
C. Find
P(X≥6)
C. Find
P(X>6)
0.983
0.931