Ch4:

4.1 Probability Density Function (pdf)

4.2 CDFs and expected values

Continuous pdfs, CDFs and Expectation Section 4.1-2

A Random Variable: is a function on the outcomes of an experiment; i.e. a function on outcomes in S.

A discrete random variable is one with a sample space that is finite or countably infinite. (Countably infinite => infinite yet can be matched to the integer line)

A continuous random variable is one with a continuous sample space.

Still going through these steps (even if it is deep in our heart):

1) Identify the experiment of interest and understand it well (including the associated population)

2) Identify the sample space (all possible outcomes)

3) Identify an appropriate random variable that reflects what you are studying (and simple events based on this random variable)

4) Construct the probability distribution associated with the simple events based on the random variable

Continuous pdfs, CDFs and Expectation Section 4.1-2

Example: Continuous data:

Grades out of 10012 77 83 87 9248 78 84 88 9455 78 84 88 9457 78 84 88 9462 79 84 89 9665 79 85 89 9767 80 85 89 9769 80 86 90 9869 80 86 90 9871 80 86 9171 80 86 9172 81 87 9173 81 87 9173 81 87 9174 82 87 9174 82 87 9176 82 87 9276 82 87 9277 82 87 92

Continuous pdfs, CDFs and Expectation Section 4.1-2

Class or Bin FrequencyRelative

Frequency10-<20 1 0.0120-<30 0 030-<40 0 040-<50 1 0.0150-<60 2 0.0260-<70 5 0.0670-<80 16 0.1980-<90 39 0.46

90-<100 21 0.25 Total 85 1

Continuous pdfs, CDFs and Expectation Section 4.1-2

Histogram of x

x

Density

20 40 60 80 100

0.00

0.01

0.02

0.03

0.04

Called density because the total area of the graph is 1.

Continuous pdfs, CDFs and Expectation Section 4.1-2

When we are looking at the entire population then,

Continuous pdfs, CDFs and Expectation Section 4.1-2

Histogram of x

x

Density

20 40 60 80 100

0.00

0.02

0.04

Histogram of x

x

Density

20 40 60 80 100

0.00

0.02

0.04

Histogram of x

x

Density

20 40 60 80 100

0.00

0.02

0.04

Histogram of x

x

Density

20 40 60 80 100

0.00

0.04

Histogram of x

x

Density

20 40 60 80 100

0.00

0.04

0.08

Histogram of x

x

Density

20 40 60 80 1000.0

0.4

0.8

Continuous pdfs, CDFs and Expectation Section 4.1-2

We can reduce the width of the class and have it approach zero.

Histogram of x

x

Density

20 40 60 80 100

0.00

0.02

0.04

0.06

0.08

0.10

Continuous pdfs, CDFs and Expectation Section 4.1-2

The Height can then be found using the function that approximates the structure of the sample space. And,

Very small close to zero

Also very small close to zero

Continuous pdfs, CDFs and Expectation Section 4.1-2

So the chance of observing the value x is pretty much zero!

This is the major difference from the discrete distribution structure, the rest follows:

Probability distributions for discrete rvs Section 3.2

For continuous random variables, we call f(x) the probability density function (pdf).

From the axioms of probability, we can show that:

1.

2.

1.

2.Opposed to

Probability distributions for discrete rvs Section 3.2

Based on the pdf we construct the CDF, F(x),

Opposed to

We can also find the pdf using the CDF if we note that:

Probability distributions for discrete rvs Section 3.2

So, for any two numbers a, b where a < b,

Expected values Section 3.3

The expected value E(X) of a continuous random variable is,

Compared to

Expected values Section 3.3

The variance, E[(X - E(X))2] of a contunuous random variable is,

It measures the amount of variation or uncertainty in the to be observed (or the observed) value of a random variable.

Expected values Section 3.3

The standard deviation,

Also measures the amount of variation or uncertainty in the to be observed (or the observed) value of a random variable. It has the same units as the random variable.

Expected values Section 3.3

Some properties of expectation:Let h(X) be a function, a and b be constants then,

Example: The current measured in a thin wire has an equal chance of being in an the interval [1, 10] milliampere. What is the probability that at a certain time the measured current is more than 7 mA’s? What is the mean current over time? The variance?

Continuous pdfs, CDFs and Expectation Section 4.1-2

1) Identify the experiment of interest and understand it well

2) Identify the sample space

3) Identify an appropriate random variable that reflects what you are studying

Continuous pdfs, CDFs and Expectation Section 4.1-2

4) Construct the probability distribution associated with the simple events based on the random variable

Continuous pdfs, CDFs and Expectation Section 4.1-2

0 2 4 6 8 10

0.00

0.02

0.04

0.06

0.08

0.10

x

density

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

x

F(x)

4) Construct the probability distribution associated with the simple events based on the random variable

Continuous pdfs, CDFs and Expectation Section 4.1-2

The Uniform distribution

Continuous pdfs, CDFs and Expectation Section 4.1-2

Example: Observing the time until a red car passes through the main and sixth intersection.

S =

Random variable X = time till we observe a red car go through main and 6th. X(0) = 0, X(0.0001) = 0.0001

A one-to-one transformation of S.

Probability distribution:

Continuous random variable! Sample space is continuous.

Continuous pdfs, CDFs and Expectation Section 4.1-2