Transcript 6.9 - Discrete Random Variables & Distributions

6.9 – Discrete Random
Variables
IBHLY2 - Santowski
(A) Random Variables
Now we wish to combine some basic statistics with
some basic probability  we are interested in the
numbers that are associated with situations resulting
from elements of chance i.e. in the values of random
variables
We also wish to know the probabilities with which
these random variables take in the range of their
possible values  i.e. their probability distributions
(A) Random Variables
So 2 definitions need to be clarified:
(i) a discrete random variable is a variable quantity which occurs
randomly in a given experiment and which can assume certain, well
defined values, usually integral  examples: number of bicycles sold in
a week, number of defective light bulbs in a shipment
discrete random variables involve a count
(ii) a continuous random variable is a variable quantity which occurs
randomly in a given experiment and which can assume all possible
values within a specified range  examples: the heights of men in a
basketball league, the volume of rainwater in a water tank in a month
continuous random variables involve a measure
(B) CLASSWORK
CLASSWORK: (to review the distinction
between the 2 types of random
variables)
Math SL text, pg 710, Chap29A, Q1,2,3
Math HL Text, p 728, Chap 30A, Q1,2,3
(C) The Distributions of Random Variables
For any random variable, there is an associated probability
distribution:
the discrete probability distribution is associated with
discrete random variables  so this probability distribution
describes a discrete random variable in terms of the
probabilities associated with each individual value that the
variable may take
the normal distribution is associated with continuous random
variables
we will initially consider only discrete data and their associated
probability distributions
(C) The Distributions of Random Variables
We will toss three coins. The random variable, X, will represent the
number of heads obtained. Construct a table and a graph to represent
the discrete probability distribution
the probability of exactly 1 head in three tosses will be written as
P(X=1) (which I can read as the probability that my random variable (#
of heads) has the value 1 i.e. 1 head is tossed)
It can be determined in many different ways  I will use binomial
probability distribution ((p + q)3) from our last section  C(3,1) x (0.5)1
x (0.5)3-1 = 3 x 0.5 x 0.25 = 0.375
Or I could use a GDC and determine binompdf(3,0.5,1) = 0.375
(Or I could use the Fundamental Counting Principle ==> p(H) x
p(T) x p(T) x C(3,1) ) = 0.375
Likewise, I could do similar calculations to find the associated
probabilities for 0,1,2,3 Heads ==> I will write this as P(X = x) and
equate it to C(3,x) x (0.5)x x (0.5)3-x, x = 0,1,2,3
(C) The Distributions of Random Variables
I get the following graph:
I get the following table:
x
P(X=x)
Binomial Probability Distribution
P(X = x)
0.125
0.4
P r o b a b ilit y
0
0.3
1
0.375
2
0.375
0.2
0.1
0
3
0.125
0
1
Number of Heads
2
3
(C) The Distributions of Random Variables
ex 2. Of the 15 light bulbs in a box, 5 are defective. Four bulbs are
chosen at random from the box. Let the random variable, X, represent
the number of defective bulbs selected. Construct a table and graph to
represent this distribution.
(NOTE: the events are NOT independent (as selecting a defective bulb
first, now influences the probabilities of the selection of a second bulb
 therefore, binompdf(4,1/3,x) will give us different answers than the
following approach:
the number of ways of selecting x defective bulbs from the 5 is C(5,x)
the number of ways of selecting (4 - x) non-defective bulbs is C(10, 4-x)
the number of ways of selecting 4 bulbs from 15 is C(15,4)
so our basic probability formula would be (# of specific events) ) (total
# of events) = [C(5,x) x C(10,4-x)] ) [C(15,4)]
(C) The Distributions of Random Variables
I get the following table:
x
P(X = x)
0
0.154
I get the following graph:
Selection of Defective Light Bulbs
1
0.440
P r o b a b ilit y
0.5
0.4
0.3
2
0.330
0.2
0.1
3
0.073
4
0.004
0
0
1
2
# of Defective Bulbs
3
4
(D) Homework
SL Math text, Chap 29B, p712, Q1-7
HL Math text, Chap 30B, p730, Q1-11