1. One Random Variable
Random Process

2. 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 variable
Consider 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

3. 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]

4. 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

5. 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

6. The Cumulative Distribution Function
The cdf has the following properties:

7. 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

8. Example 3 (Cont’d) The cdf for event can be obtained by getting first
By using property (vii)

9. 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

10. 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

11. The Probability Density Function
The probability density function (pdf) is defined as
The properties of pdf

12. The Probability Density Function

13. 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

14. Example 4 The pdf of the uniform random variable is given by
The cdf will be

15. 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

16. Pdf of Discrete Random Variable
Remember these:
Unit step function
The pdf for a discrete random variable is

17. 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

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

19. 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

20. The Expected Value of X The properties of variance
The n-th moment of the random variable is

21. Some Continuous Random Variable

22. Some Continuous Random Variable

23. Some Continuous Random Variable

24. Some Continuous Random Variable

25. Some Continuous Random Variable

26. 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

27. Transform Methods
The characteristic function of a random variable X is
The inversion formula that represent pdf is

28. Example 7: Exponential Random Variable

29. 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:

30. Example 8: Geometric Random Variable

31. Transform Methods
The moment theorem states that the moments of X are given by

32. Example 9

33. 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

34. The Laplace Transform of The Pdf
The Laplace transform of the pdf can be written as
The moment theorem also holds

35. Example 10