INTRODUCTION TO STATISTICS & PROBABILITY

Chapter 4: Probability: The Study of Randomness

(Part 3)

Dr. Nahid Sultana

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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*

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4.4 Means and Variances of Random Variables

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

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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

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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:

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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 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 222

221

21

2 )()(...)()()( σ

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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: HHH Value 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)(...22)2(1

2)1(2

====+=

+++=

−+−+−+−=

µ−++µ−+µ−= kpXkxpXxpX

xXσ

The mean µ of X , 2/3=Xμ

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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:

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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 = b2σ2

X

Rule 2: If X and Y are two independent random variables, then: µX+Y = µX + µY

σ2X+Y = σ2

X + σ2Y

Rule 3: If X and Y are not independent but have correlation ρ, then: µX+Y = µX + µY

σ2X+Y = σ2

X + σ2Y + 2ρσXσY

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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.

? ?

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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.

. of values different produce wouldsamples randomdifferent all, After? of estimate accurate an be canHow

xμx

. µparameter the to closer and closer get to guaranteed isstatistic the samples, larger and larger taking on keep weIf

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.