# Random Processes in Communication and Control/M-Sep14

## Contents

## Last Time[edit]

PMF

a)

b)

c) event

## Some Useful Random Variables[edit]

### Bernoulli R.V[edit]

success probability

#### Example[edit]

1) Flip a coin # of H

2) Manufacture a Chip # of acceptable chips

3) Bits you transmit successfully by a modem

### Geometric Random Variable[edit]

Number of trials until (and including) a success for an underlying Bernoulli

#### Example[edit]

1) Repeated coin flips # of tosses until H

2) Manufacture chips 3 of chips produced until an acceptable time

### Binomial R.V[edit]

"# of successes in n trials"

#### Example[edit]

1) Flip a coin n times. # of heads.

2) Manufacture n chips. # of acceptable chips.

Note: Binomial where are independent Bernoulli trials

Note: n=1; Binomial=Bernoulli;

### Pascal R.V[edit]

"number of trials until (and including) the kth success with an underlying Bernoulli"

where is successes in trials

Note: Pascal where are geometric R.V.

Note: K=1 Pascal=Geometric

#### Example[edit]

# of flips until the kth H

### Discrete Uniform R.V.[edit]

#### Example[edit]

1) Rolling a die.

2) Flip a fair coin. =# of H

### Poisson R.V.[edit]

(Exercise) limiting case of binomial with

PMF is a complete model for a random variable

## Cumulative Distribution Function[edit]

Like PMF, CDF is a complete description of random variable.

### Example[edit]

Flip the coins # of H

### Properties of CDF[edit]

- a)

"starts at 0 and ends at 1"

- b) For all ,

"non-decreasing in x"

- c) For all

"probabilities can be found by difference of the CDF"

- d) For all ,

"CDF is right continuous"

- e) For

"For a discrete random variable, there is a jump (discontinuity) in the CDF at each value . This jump equals

- f) for all

"Between two jumps the CDF is constant"

- g)

## Continuous Random Variables[edit]

outcomes uncountable many

### Example[edit]

T: arrival of a partical

V: voltage

: angle

: distance

No PMF,

### Theorem[edit]

For any random variable (continuous or discrete)

- a)

- b) is nondecreasing in

- c)

- d) is right continuous

### Example[edit]

where A, B are intervals of the same length contained in [0,1]

(exercise)

### Probability Density Function (PDF)[edit]

discrete: PMF <--> CDF (sum/difference)

continuous <---> (derivative/integral)

### Theorem: Properties of PDF[edit]

- a) ( is nondecreasing)

- b)

- c)

### Theorem[edit]

### Some useful continuous Random Variables[edit]

#### Uniform R.V[edit]

#### Exponential R.V[edit]

#### Gaussian (Normal) R.V.[edit]