Transcript Probability Distributions

Probability Distributions
We need to develop probabilities of all possible
distributions instead of just a particular/individual
outcome
Many probability experiments have numerical outcomes
which can be counted or measured
A random variable X has a single value for each outcome
in an experiment. Ex. If ‘X’ is the number rolled with a
die, then X has a different value for each of the 6 possible
outcomes. Random variables can be discrete or
continuous.
Discrete  values that are separate from each other
(number of possible values can be small)
Continuous  have an infinite number of possible values
in a continuous interval. This chapter involves discrete
random variables.
Classify each as either random or continuous.
- number of customers on a paper route
- Length of time it takes to deliver the papers
- amount of money that you can make delivering papers
Ask yourself if the values can be measured to an infinite
point (ie. Does it require integers or fractions of a
decimal.
Uniform Probability Distributions
This just means that we need to calculate the probability
of a group A,B,C,D or E being chosen for a particular
place.
Ex. Determine the probability distribution for comparing
the probabilities of Group A presenting their project first,
second, third, fourth or fifth.
Since each group has an equal probability for choosing
each of the five positions, each prob is 1/5.
Heading Random Variable, x Probability, P(x)
Position 1
1/5
All outcomes are equally likely in a single trial. This
distribution has a uniform probability distribution. The
sum of the probabilities must equal 1 since it includes all
possible outcomes.
For all values of x,
P(X) = 1/n, where n is the number of possible outcomes in
the experiment.
Expected values E(x) is the predicted average of all
possible outcomes of an experiment. The expectation is
equal to the sum of the products of each outcome with its
probability
E(x) = x1P(x1) + … + xnP(xn)
n
x1 P ( x1 )
= 
i 1
Consider a game in which you gain the points if it is an
even number, and if you roll an odd number you lose that
number of points.
Show the distribution of points in this game using a table.
Headings
Number on Upper Face
What is the expected outcome?
Is this a fair game? Why?
Points,x
Probability, P(X)
It is not because the points gained and lost are not equal.
For a game to be fair, the expected outcome must be 0.
A building store has different sizes of 2x4s. There are 66m boards, 5-8 m boards, 3-4m boards and 4-10m boards.
Length of board (M), x Probability, P(x)
4m
6m
8m
10m
3/18
6/18
5/18
4/18
E(x) = 4(3/18) + 6(6/18) + 8(5/18) + 10(4/18)
The expected length is 7.6444
Homework
Pg 374 # 1,3,6,8,9