Statistics, study of probability, The Mean of a Random Variable, The Variance of a Random Variable, Rules for Means and Variances, The Law of Large Numbers,

1. INTRODUCTION TO
STATISTICS & PROBABILITY
Chapter 4:
Probability: The Study of Randomness
(Part 3)
Dr. Nahid Sultana
1

2. 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*
2

3. 4.4 Means and Variances of
Random Variables
3
 The Mean of a Random Variable
 The Variance of a Random Variable
 Rules for Means and Variances
 The Law of Large Numbers

4. 4
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:
∑=++++== iikkx
pxpxpxpxpxXEμ ...)( 332211

5. 5
The Mean of a Random Variable
(Cont…)
Consider tossing a fair coin 3 times.
Define X = the number of heads
obtained.
X = 0: TTT
X = 1: HTT THT TTH
X = 2: HHT HTH THH
X = 3: HHH
Value 0 1 2 3
Probability 1/8 3/8 3/8 1/8
The mean µ of X is
5.12/38/12
)8/1*3()8/3*2()8/3*1()8/1*0(
...332211
===
+++=
++++= kkx
pxpxpxpxμ

6. 6
The Mean of a Random Variable
(Cont…)
Mean of a Continuous Random Variable
If X is a continuous random variable with probability distribution f(x)
then the mean or expected value of X is found by:
∫
∞
∞−
== dxxxfXEμx )()(
Example: Suppose we have a continuous random variable X with
probability density function given by
Calculate E(X).
Solution:

7. 7
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 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:
The standard deviation of a random variable is the square root of the variance.
∑ µ−=µ−++µ−+µ−== iXikXkXXX
pxpxpxpxXVar 22
2
2
21
2
1
2
)()(...)()()( σ

8. 8
Variance of a Random Variable
(Cont…)
Example: Consider tossing a fair coin 3 times.
Define X = the number of heads obtained.
X = 0: TTT
X = 1: HTT THT TTH
X = 2: HHT HTH THH
X = 3: HHHValue 0 1 2 3
Probability 1/8 3/8 3/8 1/8
75.04/332/24)32/12(2)32/3(2)32/9(2
8/1*4/98/3*4/18/3*4/18/1*4/9
8/1*
2
)2/33(8/3*
2
)2/32(8/3*
2
)2/31(8/1*
2
)2/30(
2
)(...2
2
)2(1
2
)1(
2
====+=
+++=
−+−+−+−=
µ−++µ−+µ−= kpXkxpXxp
X
xXσ
The mean µ of X , 2/3=Xμ

9. 9
Variance of a Random Variable
(Cont…)
Variance of a Continuous Random Variable
If X is a continuous random variable with probability distribution f(x) then
the variance of X is given by:
∫
∞
∞−
−== dxxfXXVar xX )()()( 22
µσ
Example: Suppose we have a continuous random variable X with
probability density function given by
Calculate Var(X).
Solution:
222
))(()( XEXEX
−=σTheorem:

10. 10
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
σ2
a+bX = b2σ2
X
Rule 2: If X and Y are two independent random variables, then:
µX+Y = µX + µY
σ2
X+Y = σ2
X + σ2
Y
Rule 3: If X and Y are not independent but have correlation ρ, then:
µX+Y = µX + µY
σ2
X+Y = σ2
X + σ2
Y + 2ρσXσY

11. 11
Rules for Means and Variance (Cont…)
Example:
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.
??

12. 12
The Law of Large Numbers
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.
.ofvaluesdifferentproducewouldsamplesrandom
differentall,After?ofestimateaccurateanbecanHow
x
μx
.µparameterthetocloserandclosergettoguaranteedis
statisticthesamples,largerandlargertakingonkeepweIf
x
Suppose we would like to estimate an unknown mean µ.
We could select an SRS and calculate sample mean .
However, a different SRS would probably yield a different sample mean.