6/24/2015 william b. vogt, carnegie mellon, 45-733 1 45-733: lecture 5 (chapter 5) continuous random...

04/18/23William B. Vogt, Carnegie Mellon, 45-7331

45-733: lecture 5 (chapter 5)45-733: lecture 5 (chapter 5)Continuous Random Variables


Random variableRandom variable

Is a variable which takes on different values, depending on the outcome of an experiment– X= 1 if heads, 0 if tails– Y=1 if male, 0 if female (phone survey)– Z=# of spots on face of thrown die– W=% GDP grows this year– V=hours until light bulb fails



Discrete random variable– Takes on one of a finite (or at least countable)

number of different values.– X= 1 if heads, 0 if tails– Y=1 if male, 0 if female (phone survey)– Z=# of spots on face of thrown die



Continuous random variable– Takes on one in an infinite range of different

values– W=% GDP grows this year– V=hours until light bulb fails– Particular values of continuous r.v. have 0

probability– Ranges of values have a probability


Probability distributionProbability distribution

Again, we are searching for a way to completely characterize the behavior of a random variable

Previous tools:– Probability function: P(X=x)– Cumulative probability distribution: P(Xx)


Probability functionProbability function

Continuous variables may take on any value in a continuum

So, the probability that they take on any particular value is zero:– Probability that the temp in this room is exactly

72.00534 degrees is zero– Probability that the economy will grow exactly

1.5673101% is zero P(X=x)=0 always for continuous r.v.s So, the probability function is useless for

continuous r.v.s


Cumulative distribution Cumulative distribution

However, we can still talk coherently about the cdf

There is some positive probability that the temp in the room is less than 75 degrees

There is some positive probability that the economy this year will grow by less than 1%

FX(x)=P(Xx)



We can also calculate the probability that a continuous random variable falls in a range

The probability that growth will be between 0.5% and 1%:

5.0115.0

15.05.0

"15.0""5.0"1

XPXPXP

XPXP

XXPXP




The probability that growth will be between 0.5% and 1%:

5.0115.0 XX FFXP




The probability that X will be between a and b:

aFbFbXaP XX


Density functionDensity function

Although the probability function is useless with continuous r.v.s, there is an analogue to it

The probability density function for X is a function with two properties:– fX(x)0 for each x

– The area under f between any two points a,b is equal to FX(b)- FX(a)



ba

FX(b)- FX(a)


The uniform distributionThe uniform distribution

The simplest continuous distributionThe uniform distribution assigns equal

“probability” to each value between 0 and 1



Density

0 1

1

x

x

x

xf X

10

101

00



Cdf:

0 1

1

x

xx

x

xFX

11

10

00



The probability that X is between 0.25 and 0.55: – Area=0.3*1=0.3– P(0.25 x 0.55)=0.3

0.550.25

1



The probability that X is between 0.25 and 0.55: – P(0.25 x 0.55)= FX(0.55)- FX(0.25)

– P(0.25 x 0.55)=0.55- 0.25– P(0.25 x 0.55)=0.3



The area under the whole density is 1The area to the left of any point, x, is

– P(Xx)

– FX(x)



0 1

1

Area=1

0 1

1

Area=FX(x)

x


Expected valueExpected value

The expected value of a random variable is its “average”

Imagine taking N independent draws on a random variable X– Calculate the mean of the N draws– Now imagine N going to infinity– The mean as N goes to infinity is the expected

value of XExpected value of X is written E(X)



The expected value of a random variable is its “average”

Imagine taking N independent draws on a random variable X– Calculate the mean of the N draws– Now imagine N going to infinity– The mean as N goes to infinity is the expected

value of X



Expected value of X is written E(X):

2

222

22

XX

X

X

X

XEXE

XEXE

XE



There are addition rules for expectations in continuous variables, just as in discrete

If Z is a random variable defined by Z=a+bX, where a,b are constants (non-random)

XVbbXaVZV

XbEabXaEZE

2



Often, we use these rules to standardize a random variable

To standardize a random variable means to make its mean 0 and its variance 1:

1 varianceand 0mean has Z

X

XEXZ


Density, expectation, varianceDensity, expectation, variance

Mean is about where the middle of the density is:

E(X) E(Y)

fX

fY



Variance is about how spread out the density is:

fX

fY YVXV



Consider two uniformly distributed random variables:

x

x

x

yfY

10

15.02

5.00

x

x

x

xf X

10

101

00



Consider two uniformly distributed random variables:

0 1

1

2

fX

fY


Expectation, varianceExpectation, variance

Example (Problem 10, pg 194):– A homeowner installs a new furnace– New furnace will save $X in each year (a

random variable)– Mean of X is 200, standard deviation 60– The furnace cost $800, installed– What are the savings over 5 years (ignoring

the time value of money), in expectation and variance?


Expectation, varianceExpectation, variance

Example (Problem 10, pg 194):– Savings = X1+ X2+ X3+ X4+ X5-800

– E(Savings)= E(X1)+ E( X2)+ E( X3)+ E( X4)+ E( X5)-800=5*200-800=200

– V(Savings)= V(X1)+ V( X2)+ V( X3)+ V( X4)+ V( X5)=5*(60)2=18000

Assuming independence

savings=1800=134

6/24/2015 william b. vogt, carnegie mellon, 45-733 1 45-733: lecture 5 (chapter 5) continuous random...

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