one random variable random process. the cumulative distribution function we have already known that...
TRANSCRIPT
![Page 1: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/1.jpg)
One Random Variable
Random Process
![Page 2: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/2.jpg)
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
![Page 3: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/3.jpg)
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]
![Page 4: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/4.jpg)
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
![Page 5: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/5.jpg)
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
![Page 6: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/6.jpg)
The Cumulative Distribution Function The cdf has the following properties:
![Page 7: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/7.jpg)
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
![Page 8: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/8.jpg)
Example 3 (Cont’d) The cdf for event can be
obtained by getting first
By using property (vii)
![Page 9: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/9.jpg)
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
![Page 10: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/10.jpg)
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
![Page 11: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/11.jpg)
The Probability Density Function The probability density function (pdf) is
defined as
The properties of pdf
![Page 12: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/12.jpg)
The Probability Density Function
![Page 13: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/13.jpg)
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
![Page 14: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/14.jpg)
Example 4 The pdf of the uniform random variable is
given by
The cdf will be
![Page 15: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/15.jpg)
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
![Page 16: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/16.jpg)
Pdf of Discrete Random Variable Remember these:
Unit step function
The pdf for a discrete random variable is
![Page 17: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/17.jpg)
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
![Page 18: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/18.jpg)
Conditional Cdf’s and Pdf’s The conditional cdf of X given C is
The conditional pdf of X given C is
![Page 19: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/19.jpg)
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
![Page 20: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/20.jpg)
The Expected Value of X The properties of variance
The n-th moment of the random variable is
![Page 21: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/21.jpg)
Some Continuous Random Variable
![Page 22: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/22.jpg)
Some Continuous Random Variable
![Page 23: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/23.jpg)
Some Continuous Random Variable
![Page 24: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/24.jpg)
Some Continuous Random Variable
![Page 25: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/25.jpg)
Some Continuous Random Variable
![Page 26: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/26.jpg)
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
![Page 27: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/27.jpg)
Transform Methods The characteristic function of a random
variable X is
The inversion formula that represent pdf is
![Page 28: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/28.jpg)
Example 7: Exponential Random Variable
![Page 29: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/29.jpg)
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:
![Page 30: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/30.jpg)
Example 8: Geometric Random Variable
![Page 31: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/31.jpg)
Transform Methods The moment theorem states that the
moments of X are given by
![Page 32: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/32.jpg)
Example 9
![Page 33: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/33.jpg)
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
![Page 34: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/34.jpg)
The Laplace Transform of The Pdf The Laplace transform of the pdf can be
written as
The moment theorem also holds
![Page 35: One Random Variable Random Process. The Cumulative Distribution Function We have already known that the probability mass function of a discrete random](https://reader036.vdocuments.site/reader036/viewer/2022062309/56649f1b5503460f94c30a66/html5/thumbnails/35.jpg)
Example 10