Random Process-M1-Probability Theory and Random Processes

The study of probabilities originally came from -------

gambling

What are Expectation of Product of random variables?

If X and Y are mutually independent random variables, then the expectation of their product exists and is E(XY) = E(X) E(Y)

Give Examples for Probability?

1.If the experiment consists of flipping two coins, then the sample space is: S = {(H, H), (H, T), (T, H), (T, T)} 2.If the experiment consists of tossing two dice, then the sample space is: S = {(i, j) | i, j = 1, 2, 3, 4, 5, 6}

What is Probability theory ?

It is a study of random or unpredictable experiments and is helpful in investigating the important features of these random experiments

Give Eg for Random Variable?

Suppose that we toss two coins and consider the sample Space associated with this experiment. Then S={HH,HT,TH,TT}. Define the random variable X as follows: X is the number of heads obtained in the two tosses. Hence X(HH) = 2, X(HT) = 1 = X(TH) & X(TT) = 0. Note that to every s in S there corresponds exactly one value X(s). Different values of x may lead to the same value of S. Eg. X(HT) = X(TH)

What are objectives of learning Probability Theory and Random Processes?

To study ● Probability: its applications in studying the outcomes of random experiments ● Random variables: types, characteristics, modeling random data ● Stochastic systems: their reliability ● Random Processes: types, properties and characteristics with special reference to signal processing and trunking theory. Learning Outcomes :students will be able to (i)model real life random processes using appropriate statistical distributions; (ii) compute the reliability of different stochastic systems; (iii) apply the knowledge of random processes in signal processing and trunking theory.

Explain Independent Events ?

Two events E and F are independent if p(EF)=p(E)p(F) Two events are not independent are said to be dependent. p(EF) = p(E)p(F) if and only if p(E|F) = p(E). If E and F are independent, then so are E and Fc.

Explain Probability Definitions?

•A subset (say E) of the sample space is called an event. In other words, events are sets of outcomes. • ( If the outcome of the experiment is contained in E, then we say E has occurred.) •For each event, we assign a number between 0 and 1, which is the probability that the event occurs. Eg: 1.If the experiment consists of flipping two coins, and E is the event that a head appears on the first coin, then E is: E = {(H, H), (H, T)} 2.If the experiment consists of tossing two dice, and E is the event that the sum of the two dice equals 7, then E is: E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

what is Random Experiments ?

•An experiment whose outcome or result can be predicted with certainty is called a deterministic experiment. Although all possible outcomes of an experiment may be known in advance, the outcome of a particular performance of the experiment cannot be predicted owing to a number of unknown causes. Such an experiment is called a random experiment •A random experiment is an experiment that can be repeated over and over, giving different results.•e.g A fair 6-faced cubic die, the no. of telephone calls received in a board in a 5-min. interval.

Give Coin toss Example of Probability?

•Consider an experiment in which a coin is tossed twice. •Sample space: { HH, HT, TH, TT } •Let E be the event that at least one head shows up on the two tosses. Then E = { HH, HT, TH } •Let F be the event that heads occurs on the first toss. Then F = { HH, HT } •A natural assumption is that all four possible events in the sample space are equally likely, i.e., each has probability ¼. Then the P(E) = ¾ and P(F) = ½.

Define Probability?

•For discrete math, we focus on the discrete version of probabilities. •For each random experiment, there is assumed to be a finite set of discrete possible results, called outcomes. Each time the experiment is run, one outcome occurs. The set of all possible outcomes is called the sample space.

Give Example of probability based on text definition?

•Rolling a die is a random experiment. •The outcomes are: 1, 2, 3, 4, 5, and 6, presumably each having an equal probability of occurrence (1/6). •One event is "odd numbers", which consists of outcomes 1, 3, and 5. The probability of this event is 1/6 + 1/6 + 1/6 = 3/6 = 0.5. :"

Give Example for Probability?

•Rolling a die is a random experiment. •The outcomes are: 1, 2, 3, 4, 5, and 6. Suppose the die is "loaded" so that 3 appears twice as often as every other number. All other numbers are equally likely. Then to figure out the probabilities, we need to solve: p(1) + p(2) + p(3) + p(4) + p(5) + p(6) = 1 and p(3) = 2*p(1) and p(1) = p(2) = p(4) = p(5) = p(6). Solving, we get p(1) = p(2) = p(4) = p(5) = p(6) = 1/7 and p(3) = 2/7. •One event is "odd numbers", which consists of outcomes 1, 3, and 5. The probability of this event is: p(odd) = p(1) + p(3) + p(5) = 1/7 + 2/7 + 1/7 = 4/7.

Why are Probabilities Important?

•They help you to make good decisions, e.g.,-Decision theory •They help you to minimize risk, e.g.,-Insurance • They are used in average-case time complexity analyses of - Computer algorithms. • They are used to model processes in- Engineering.