random signals , significance of random signals in probability theory? , What is sample space?

SHORT QUESTIONS WITH ANSWERS
Q.1. What are random signals? What is significance of random signals in probability theory?
Ans. There is one other class of signals, the behaviour of which cannot be predicted. Such type of signals are called random signals. These signals are called random signals because the precise value of these signals cannot be predicted in advance before they actually occur. The examples of random signals are the noise interferences in communication systems. This means that the noise interference during transmission is totally unpredictable. In the same way, the noise generated by the receiver itself is random. Even some other signals which are not noise signals are also random signals. These signals cannot be modelled mathematically. Actually the electro magnetic interference is the major source of random noise.
Q.2. What do you mean by Experiment in probability theory?
Ans. An experiment is defined as the process which is conducted to get some results. If the same experiment is performed repeatedly under the same conditions, similar results are expected. But there are few experiments which do not produce the same result as we have stated above. The theory of probability mainly deals with such type of experiments. In fact, the theory of probability is applied to such type of experiments to predict the possibility of a particular output. An experiment is sometimes called trial. As an example, throw of a coin is an experiment or trial. This trial results in two outcomes namely Head and Tail.
Q.3. What is sample space?
Ans. A set of all possible outcomes of an experiment or trial is called the sample space of that experiment. It is generally denoted by ‘S’. The total number of outcomes in a sample space is denoted by n(s). As an example, the tossing of a coin has two outcomes. Hence, we may write its sample space as:
S = {H, T}
where,                            H ” Head and T ” Tail
Q.4. What is an in probability theory?
Ans. The expected subset of the sample space or happening is called an event. As an example, let us consider an experiment of throwing a cubic die. In this case, the sample space S will be as
S = {1, 2, 3, 4, 5, 6}
Now, if we want the number ‘3’ to be an outcome or an even number, i.e., {2, 4, 6}, then this subset is called an event. This is denoted by letter ‘ E’ . Hence event E is a subset of the sample space ‘S. If event E has only one outcome, then it is called an elementary event. On the other hand, if event E does not contain any out come, then it is called a null event. If E = S, then an event contains all the outcomes. Such as event is called a certain event. It always occurs, no matter what so ever is the outcome.
Q.5. What do you mean by probability? Explain.
Ans. Probability may be defined as the study of random experiments. In any random experiment, there is always an uncertainty that a particular event will occur or not. As a measure of probability of occurrence of an event, a number between 0 to 1 is assigned. If it is sure that an event will occur, then we can say that its Probability is 100% or 1. If it is not sure that an event will occur, then
we can say that its probability is 0% or 0. If it is not sure whether the event will occur or not, then its probability is between 0 and 1.
 
Q.6. Write properties of probability? Explain.
Ans. Property 1: The probability of a certain event is unity i.e.,
P (A) = 1 …(2.1)
Property 2: The probability of any event is always less than or equal to 1 and non-negative. Mathematically,
0 < P (A) < 1
Property 3 : If A and B are two mutually exclusive events, then
P(A+B) = P()A) + P (B)
Property 4: If A is any event, then the probability of not happening of A is
P (A) = 1- P (A)
where  A represents the complement of event. A
 
Property 5 : If A and B are any two events (no mutually exclusive events), then
P (A+B) = P(A) + P (B) – P (AB)
where P (AB) is called the probability of events A and B both occurring simultaneously Such an event is called joint event of A and B, and the probability P (AB) is called the joint probability.
Now, if events A and B are mutually exclusive, then the joint probability, P (AB) = 0.
Q.7. What is conditional probability?
Ans. The concept of conditional probability is used in conditional occurrences of the events. Let us consider an experiment which involves two events A and B. Now the probability of event B, given that event A has occurred, is represented by P (B/A). Similarly, P (A/B) represents probability of event A given that event B has already occurred. Therefore, P (B/A) and P (A/B) are called conditional probabilities.
Q.B. Explain random variable?
Ans. A function which can take on any value from the sample space and its range is some set of real numbers is called a random variable of the experiment. Random variables are denoted by upper case letters such as X, Y etc. and the values taken by them are denoted by lower case letters with subscripts such as x_l, x₂, y₁, y₂ etc.
Random variables may be classified as under:

1. Discrete random variables
2. Continuous random variables.

Q.9. What is a continuous random variable?
Ans. A random variable that takes on an infinite number of values is called a continuous random variable. Actually, there are several physical system (experiments) that generate continuous outputs or outcomes. Such systems generate infinte number of outputs or outcomes within the finite period. Continuous random variables may be used to define the outputs of such systems.
As an example, the noise voltage generated by an electronic amplifier has a continuous amplitude. This means that sample space S of the noise voltage
amplitude is continuous. Therefore, in this case, the random variable X has a continuous range of values.
Q.10. What do you mean by probability density function (PDF)?
Ans. The derivative of cumulative distribution function (CDF) with respect to some dummy variable is known as Probability Density Function (PDF). Probability density function (PDF) is generally denoted by f_x(x). Mathematically, PDF may he expressed as
EQUATION
 where x is a dummy variable.
Probability density function (PDF) is the more convenient representation for continuous random variable.
Q.11. What do you mean by conditional probability density function?
Ans. Out of the two random variables, one variable may take a fixed value. In this case, the PDF is called conditional. As an example, out of the two continuous random variables X and Y, let X = x.
Then we may find the conditional, PDF of Y given that X = x as,
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EQUATION