Transcript Chapter5, Sections 2 and 3

Chapter 5
Probability Distributions
Random variables
E.g., X is the number of heads obtained in 3 tosses of a coin.
[X=0] = {TTT}
[X=1] = {HTT, THT, TTH}
[X=2] = {HHT, HTH, THH}
[X=3] = {HHH}
The events corresponding to the distinct values of X are
incompatible.
The union of these events is the entire sample space.
A random variable can be discrete or continuous.
Probability distribution
Probability histogram
Figure 5.1 (p. 176)
The probability histogram of X, the number of heads in three tosses of a coin.
Example:
Two dice are tossed. The possible outcomes are shown below:
Probability distribution
for the two dice example
• The event that the sum of face values is 7:
{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}. Thus P(sum=7) = 6/36
• Similarly,
P(sum=6) = 5/36
P(sum=8) = 5/36
P(sum=5) = 4/36
P(sum=9) = 4/36
P(sum=4) = 3/36
P(sum=10) = 3/36
P(sum=3) = 2/36
P(sum=11) = 2/36
P(sum=2) = 1/36
P(sum=12) = 1/36
• This is the probability distribution for random variable X,
the sum of face values of two dice.
• How can we use it in practice? See next slide.
Want to win Monopoly Game?
Learn Counting and Probability
• Your opponent’s token is in one of the squares
• His turn consists of rolling two dice and moving the token
clockwise on the board the number of squares indicated by
the sum of dice values
• When his token lands on a property
that is owned by you, you collect rent
• It is more advantageous to have houses
or hotels on your properties because rents
are much higher than for unimproved properties.
• You might build houses or hotels on
your properties before your opponent rolls the dice
• Suppose you own most of the squares following (clockwise)
your opponent’s token.
In which square should you build houses or hotels?
Probability distribution for news
source preference
In a university, 30% of students prefer the Internet to TV for
getting news.
Four students are randomly selected. Let X be the number of
students sampled that prefer news from the Internet.
Obtain the probability distribution of X.
For one student, P(I)=0.3, P(T)=0.7. The observations on 4 students can
be considered independent.
Thus, P(TTTT) = .7 × .7 × .7 × .7 = .2401 and P(X=0) = .2401
Similarly, P(TITT) = .7 × .3 × .7 × .7 = .1029
P(X=1) = 4 × .73 × .3 = .4116
P(X=2) = 6 × .72 × .32 = .2646
P(X=3) = 4 × .7 × .33 = .0756
P(X=4) = .34 = .0081
Important distinction between a relative frequency
distribution and the probability distribution:
• A relative frequency distribution is a samplebased entity and is therefore susceptible to
variation on different occasions of sampling.
• The probability distribution is a stable entity
that refers to the entire population. It is a
theoretical structure that serves as a model for
describing the variation in the population.
The probability distribution of X can be used to calculate the
probabilities of events defined in terms of X.
Ex.: For the distribution below, P[X ≤ 1] = f(0) + f(1)
=.02 + .23 = .25