One Random Variable

One Random Variable

Random Process

The Cumulative Distribution Function We have already known that the probability mass

function of a discrete random variable is The cumulative distribution function is an alternative

approach, that is The most important thing is that the cumulative

distribution function is not limited to discrete random variables, it applies to all types of random variables

Formal definition of random variableConsider a random experiment with sample space S and event class F. A random variable X is a function from the sample space S to R with the property that the set

is in F for every b in R

X b

X b

:bA X b

The Cumulative Distribution Function The cumulative distribution function (cdf) of a

random variable X is defined as

The cdf is a convenient way of specifying the probability of all semi-infinite intervals of the real line (-∞, b]

Example 1 From last lecture’s example we know that the

number of heads in three tosses of a fair coin takes the values of 0, 1, 2, and 3 with probabilities of 1/8, 3/8, 3/8, and 1/8 respectively

The cdf is the sum of the probabilities of the outcomes from {0, 1, 2, 3} that are less than or equal to x

Example 2 The waiting time X of a costumer at a taxi

stand is zero if the costumer finds a taxi parked at the stand

It is a uniformly distributed random length of time in the interval [0, 1] hours if no taxi is found upon arrival

Assume that the probability that a taxi is at the stand when the costumer arrives is p

The cdf can be obtained as follows

The Cumulative Distribution Function The cdf has the following properties:

Example 3 Let X be the number of heads in three tosses

of a fair coin The probability of event can

be obtained by using property (vi)

The probability of event can be obtained by realizing that the cdf is continuous at and

Example 3 (Cont’d) The cdf for event can be

obtained by getting first

By using property (vii)

Types of Random Variable Discrete random variables: have a cdf that is a

right-continuous staircase function of x, with jumps at a countable set of points

Continuous random variable: a random variable whose cdf is continuous everywhere, and sufficiently smooth that it can be written as an integral of some nonnegative function

Types of Random Variable Random variable of mixed type: random

variable with a cdf that has jumps on a countable set of points, but also increases continuously over ar least one interval of values of x

where , is the cdf of a discrete random variable, and is the cdf of a continuous random variable

The Probability Density Function The probability density function (pdf) is

defined as

The properties of pdf

The Probability Density Function

The Probability Density Function A valid pdf can be formed from any

nonnegative, piecewise continuous function that has a finite integral

If , the function will be normalized

Example 4 The pdf of the uniform random variable is

given by

The cdf will be

Example 5 The pdf of the samples of the amplitude of

speech waveform is decaying exponentially at a rate α

In general we define it as

The constant, c can be determined by using normalization condition as follows

Therefore, we have We can also find

Pdf of Discrete Random Variable Remember these:

Unit step function

The pdf for a discrete random variable is

Example 6 Let X be the number of head in three coin

tosses The cdf of X is

Thus, the pdf is

We can also find several probabilities as follows

Conditional Cdf’s and Pdf’s The conditional cdf of X given C is

The conditional pdf of X given C is

The Expected Value of X The expected value or mean of a random

variable X is

Let Y = g(X), then the expected value of Y is

The variance and standard deviation of the random variable X are

The Expected Value of X The properties of variance

The n-th moment of the random variable is

Some Continuous Random Variable

Some Continuous Random Variable

Some Continuous Random Variable

Some Continuous Random Variable

Some Continuous Random Variable

Transform Methods Remember that when we perform convolution

between two continuous signal , we can perform it in another way

First we do transformation (that is, Fourier transform), so that we have

1 2f t f t

1 2 1 2F f t f t F F

Transform Methods The characteristic function of a random

variable X is

The inversion formula that represent pdf is

Example 7: Exponential Random Variable

Transform Methods If we subtitute into the

formula of yields

When the random variables are integer-valued, the characteristic function is called Fourier transform of the sequence as follows

The inverse:

Example 8: Geometric Random Variable

Transform Methods The moment theorem states that the

moments of X are given by

Example 9

The Probability Generating Function The probability generating function of a

nonnegative integer-valued random variable N is defined by

The pmf of N is given by

The Laplace Transform of The Pdf The Laplace transform of the pdf can be

written as

The moment theorem also holds

Example 10