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Discrete Random Variables
• A random variable (r.v.) assigns a numerical value to the
outcomes in the sample space of a random phenomenon.
• A discrete r.v X has a finite number of possible values. The
probability distribution of X lists the values xi and their
probabilities pi. Every pi is a number between 0 and 1. The sum
of the pi’s must equal 1.
• Examples
1. Consider the experiment of tossing a coin. Define a random
variable as follows: X = 1 if a H comes up
= 0 if a T comes up.
This is an example of a Bernoulli r.v. The Probability function
of X is given in the following table
x
P(X = x)
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0
p
1- p
1
2.
3.
4.
Let X be a r.v counting the number of girls in a family with 3
children.
The probability function of X is given in the following table.
x
P(X = x)
0
1
2
3
(0.5)3 = 0.125
3·(0.5)3 = 0.375
3·(0.5)3 = 0.375
(0.5)3 = 0.125
Toss a coin 4 times. Let X be the number of H’s. Find the
probability function of X. Draw a probability histogram.
Toss a coin until the 1st H. Let X be the number of T’s before
the 1st H. Find the probability function of X.
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Continuous random variables
•
•
•
•
A continuous r. v. X takes all values in an interval of numbers.
The probability distribution of X is described by a density curve.
The total area under a density curve is 1.
The probability of any event is the area under the density curve and above
the value of X that make up the event.
• Example
The density function of a continuous r. v. X is given in the graph below.
Find i) P(X < 7)
ii) P(6 < X < 8)
iii) P(X = 7)
iv) P(5.5 < X < 7 or 8 < X < 9)
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Normal distributions
The density curves that are most familiar to us are the normal
curves.
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Mean (expected value) of a discrete r. v.
• The mean of a r. v. X is denoted by μx and can be found using
the following formula:
 X   x  P X  x    xi  pi
x
i
• Examples:
1. The mean of the Bernoulli r.v defined in example 1 above is :
μx= 0·(1-p) + 1·p = p
2. The mean number of girls in a family with 3 children is 1.5.
• Exercise: Find the mean of X in example 3 above.
• Exercise: Read Example 4.20 on p291 in IPS.
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Example – Discrete Uniform r.v
• Roll a six-sided die. Define a r. v. X to be the number shown on
the die. That is, X = 1 if die lands on 1,
X = 2 if die lands on 2, etc.
The probability distribution of X is given in the table below:
x
P(X = x)
1
2
3
4
5
6
1/6
1/6
1/6
1/6
1/6
1/6
The mean of X is
μX = 1·(1/6) + 2·(1/6) + 3·(1/6) + 4·(1/6) +5·(1/6) + 6·(1/6) = 21/6 = 3.5 .
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Law of large numbers
• If independent observations are drawn from a population with
a finite mean , the population mean  can be estimated with
a specified degree of accuracy by the sample mean x , using
sufficiently large sample.
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Rules for Mean of r.v
•
1.
2.
3.
4.
For any two r.v’s X and Y and constants a and b,
μ x + a = μx + a .
μ b·x = b·μx .
μ b·x + a = b·μx + a .
μ X + Y = μX + μY.
•
Example:
The price X of Nike sports shoes is a random variable with mean μx = 200$.
Before the holidays Nike company had a promotion: “ Pay 10$ less for each
item and get 20% discount from the original price”.
a)
What is the mean price during the promotion.
b)
Suppose in addition that during the promotion the mean price for Nike
socks is μY = 20$. What is the expected value of your expenses if you are
to buy one pair of shoes and one pair of socks ?
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The variance of a r. v.
•
•
The variance of a r. v. is an average of the squared deviations
from the mean, (X – μx)2 .
The Variance of a discrete r. v. is


 X2   x   X 2  P X  x    x 2  P X  x    X 2

x
x

•
The standard deviation σX of a r. v. is the square root of its
variance.
• Examples:
1. The variance of a Bernoulli r.v is
σ2x = p – p2 = p(1- p)
2. The variance of the Uniform example above is
σ2x = (91/6) – (3.5)2 = 2.9167
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Rules for variances
• If X is a r. v. and a and b are constants, then
1.  X2  a   X2
2
2
2
2.  bX  b   X
2
2
2


b


3. bX  a
X
4. If X and Y are independent r. v’s then,
 X2 Y   X2   Y2
 X2 Y   X2   Y2
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•
Two random variables X and Y are independent if knowing
that any event involving X alone did or did not occur tells us
nothing about the occurrence of any event involving Y alone.
•
Example:
Consider again the Nike example above. If the stdev. of X is
σx = 10$ and the stdev. of Y is σY = 8$.
a) What is the stdev. of the shoes price during the promotion? (8).
b) What is the stdev. of your expenses if you were to buy one pair
of shoes and one pair of socks ? (12.806).
•
Example 4.34 on page 283 in IPS.
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