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Chapter 4:
Probability: The Study of Randomness
(Part 3)
Dr. Nahid Sultana
Chapter 4
Probability: The Study of Randomness
4.1 Randomness
4.2 Probability Models
4.3 Random Variables
4.4 Means and Variances of Random Variables
4.5 General Probability Rules*
4.4 Means and Variances of Random
Variables
 The Mean of a Random Variable
 The Variance of a Random Variable
 Rules for Means and Variances
 The Law of Large Numbers
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The Mean of a Random Variable
 The mean
of a set of observations is their arithmetic average.
 The mean µ of a random variable X (also called expected value of X) is
the weighted average of the possible values of X, reflecting that all
outcomes might not be equally likely.
Mean of a Discrete Random Variable
Suppose that X is a discrete random variable whose probability distribution
The mean of X is found by multiplying each possible value of X by its
probability, then adding all the products:
μ x = E ( X ) = x1 p1 + x2 p2 + x3 p3 + ... + xk pk
= ∑ xi pi
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Example: Mean of a discrete random variable
Consider tossing a fair coin 3 times.
Define X = the number of heads obtained.
Value
0
1
2
3
Probability
1/8
3/8
3/8
1/8
X = 0: TTT
X = 1: HTT THT TTH
X = 2: HHT HTH THH
X = 3: HHH
The mean µ of X is
μ x = x1 p1 + x2 p2 + x3 p3 + ... + xk pk
= (0 *1 / 8) + (1* 3 / 8) + (2 * 3 / 8) + (3 *1 / 8)
= 12 / 8 = 3 / 2 = 1.5
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The Mean of a Random Variable
Mean of a Continuous Random Variable
If X is a continuous random variable probability distribution f(x) then the
mean or expected value of X is found by:
μx = E ( X ) =
∞
∫ xf ( x)dx
−∞
Example: Suppose we have a continuous random variable X with
probability density function given by
Calculate E(X).
Solution:
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Variance of a Random Variable
Since we use the mean as the measure of center for a discrete random
variable, we’ll use the standard deviation as our measure of spread.
Variance of a Discrete Random Variable
Suppose that X is a discrete random variable whose probability distribution is:
and that µX is the mean of X. The variance of X is found by multiplying each
squared deviation of X by its probability and then adding all the products:
Var ( X ) = σ X2 = ( x1 − µ X ) 2 p1 + ( x2 − µ X ) 2 p2 + ... + ( xk − µ X ) 2 pk
= ∑ ( xi − µ X ) 2 pi
The standard deviation of a random variable is the square root of the variance.
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Example: Variance of a discrete random variable
Consider tossing a fair coin 3 times.
Define X = the number of heads obtained.
Value
0
1
2
3
Probability
1/8
3/8
3/8
1/8
The mean µ of X is
2
2
X = 0: TTT
X = 1: HTT THT TTH
X = 2: HHT HTH THH
X = 3: HHH
μX = 3 / 2
2
2
σ X = ( x1 − µ ) p1 + ( x2 − µ X ) p2 + ... + ( xk − µ X ) pk
X
2
2
2
2
= (0 − 3 / 2) * 1 / 8 + (1 − 3 / 2) * 3 / 8 + ( 2 − 3 / 2) * 3 / 8 + (3 − 3 / 2) * 1 / 8
= 9 / 4 *1 / 8 + 1 / 4 * 3 / 8 + 1 / 4 * 3 / 8 + 9 / 4 *1 / 8
= 2(9 / 32) + 2(3 / 32) = 2(12 / 32) = 24 / 32 = 3 / 4 = 0.75
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The Variance of a Random Variable
Variance of a Continuous Random Variable
If X is a continuous random variable probability distribution f(x) then the
variance of X is given by:
∞
Var ( X ) = σ X = ∫ ( X − µ x ) f ( x)dx
2
2
−∞
Theorem:
σ X = E( X 2 ) − E( X )
2
Example: Suppose we have a continuous random variable X with
probability density function given by
Calculate Var(X).
Solution:
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Rules for Means and Variance
Rules for Means and Variance
Rule 1: If X is a random variable and a and b are fixed numbers, then:
µa+bX = a + bµX
σ2a+bX = b2σ2X
Rule 2: If X and Y are two independent random variables, then:
µX+Y = µX + µY
σ2X+Y = σ2X + σ2Y
Rule 3: If X and Y are not independent but have correlation ρ, then:
µX+Y = µX + µY
σ2X+Y = σ2X + σ2Y + 2ρσXσY
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Example: Investment. You invest 20% of your funds in Treasury bills and
80% in an “index fund” that represents all U.S. common stocks. Your rate of
return of over time is the proportional to that of the T-bills (X) and of the index
fund (Y), such that
R = 0.2 X + 0.8 Y.
The Law of Large Numbers
Suppose we would like to estimate an unknown µ.
We could select an SRS and calculate sample mean
.
However, a different SRS would probably yield a different sample mean.
How can x be an accurate estimate of μ? After all, different
random samples would produce different values of x.
If we keep on taking larger and larger samples, the statistic
x is guaranteed to get closer and closer to the parameter µ .
The law of large numbers says that as the number of observations
drawn increases, the sample mean of the observed values gets closer
and closer to the mean µ of the population.
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