1.Define Probability.

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Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen. The meaning of probability is basically the extent to which something is likely to happen.

2.Define Random Signal.

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Signals can be divided into two main categories - deterministic and random. The term random signal is used primarily to denote signals, which have a random in its nature source.
As an example we can mention the thermal noise, which is created by the random movement of electrons in an electric conductor.
Apart from this, the term random signal is used also for signals falling into other categories, such as periodic signals, which have one or several parameters that have appropriate random behavior. An example is a periodic sinusoidal signal with a random phase or amplitude

3.What are random signals? What is significance of random signals in probability theory?

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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 interference 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 electromagnetic interference is the major source of random noise.

4.What is an probability theory?

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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.

5.What do you mean by Experiment in probability theory?

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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. 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.

6. What Is Conditional Probability?

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Conditional probability is defined as the likelihood of an event or outcome occurring, based on the occurrence of a previous event or outcome. Conditional probability is calculated by multiplying the probability of the preceding event by the updated probability of the succeeding, or conditional, event.

7. What is a continuous random variable?

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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 infinite 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

8.What Is a Joint Probability?

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Joint probability is a statistical measure that calculates the likelihood of two events occurring together and at the same point in time. Joint probability is the probability of event Y occurring at the same time that event X occurs.
The Formula for Joint Probability Is
Notation for joint probability can take a few different forms. The following formula
represents the probability of events intersection:
P (X⋂Y)
Where : X,Y=Two different events that intersect
P(X and Y),P(XY)=The joint probability of X and Y

9. What are Independent Events?

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In Probability, the set of outcomes of an experiment is called events. There are different types of events such as independent events, dependent events, mutually exclusive events, and so on.
If the probability of occurrence of an event A is not affected by the occurrence of another event B, then A and B are said to be independent events.

10.Write properties of probability? Explain.

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Property 1: The probability of a certain event is unity i.e.,
P (A) = 1
Property 2: The probability of any event is always less than or equal to 1 and nonnegative. Mathematically,
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 (Ā) = 1 – P (A)
where Ā represents the complement of event A.
Property 5: If A and B are any two events (not 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.

11.What is the Bayes’ Theorem ? Also computes its formula.

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In statistics and probability theory, the Bayes’ theorem (also known as the Bayes’ rule) is a mathematical formula used to determine the conditional probability of events. Essentially, the Bayes’ theorem describes the probability of an event based on prior knowledge of the conditions that might be relevant to the event.
The Bayes’ theorem is expressed in the following formula:

Where:

P(A|B) – the probability of event A occurring, given event B has occurred
P(B|A) – the probability of event B occurring, given event A has occurred
P(A) – the probability of event A
P(B) – the probability of event B

1.What is a Random Variable?

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Requirements engineering refers to the process of defining, documenting, and maintaining The values which vary without following any pattern, that is they change randomly. For example, if any experiment (flipping of a coin, or rolling of a die) has the outcome bounded to be from a given set of values, and is not fixed, the result will change every time the experiment is conducted. Such an outcome is termed as Random Variable.

2. What does Probability Distribution mean?

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The possible outcomes of an experiment have varied chances. Understanding this distribution of chances/probabilities among the possible outcomes is known as Probability Distribution.

3. List out a few examples of Discrete Probability Distribution?

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• The Bernoulli Distribution
• The Binomial Distribution
• The Geometric Distribution
• The Hypergeometric Distribution
• The Multinomial Distribution
• The Poisson Distribution
• The Negative Binomial Distribution
• Discrete uniform distribution

4. Describe the types of Probability Distributions?

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Probability is calculated for the possible outcomes of an experiment. Data can be classified as Discrete and Continuous. The probability distribution is broadly classified into two types based on the type of Data that we are working with. Continuous probability distribution and Discrete probability distribution are two types of techniques.

5.Define Probability mass function and cumulative distribution function.

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Since random variables simply assign values to outcomes in a sample space and we have defined probability measures on sample spaces, we can also talk about probabilities for random variables. Specifically, we can compute the probability that a discrete random variable equals a specific value (probability mass function) and the probability that a random variable is less than or equal to a specific value (cumulative distribution function).

6.What is the Poisson Distribution, Binomial Distribution, Rayleigh Distribution?

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A Poisson distribution is a tool that helps to predict the probability of certain events happening when you know how often the event has occurred. It gives us
the probability of a given number of events happening in a fixed interval of time.
Poisson distributions, valid only for integers on the horizontal axis. λ (also written as μ) is the expected number of event occurrences.
A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail

A Binomial Distribution shows either (S)uccess or (F)ailure.
The Rayleigh distribution is a continuous probability distribution named after the English Lord Rayleigh. It is a special case of the Weibull distribution with a scale parameter of 2. When a Rayleigh is set with a shape parameter (σ) of 1, it is equal to a chi square distribution with 2 degrees of freedom.
The notation X Rayleigh(σ) means that the random variable X has a Rayleigh
distribution with shape parameter σ. The probability density function (X > 0) is:

Where e is Euler’s number. As the shape parameter increases, the distribution gets wider and flatter.

Rayleigh dist. showing several different shape parameters, σ.

7.Give two properties of probability distribution function

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If (x1 P{x1

8..List out a few examples of Continuous Probability Distribution?

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• Normal Distribution
• Chi-Square Distribution
• Fishers F Distribution
• Students t Distribution
• The Gamma Distribution
• The Exponential Distribution
• The Beta Distribution
• The Weibull Distribution

9.Define Continuous, Discrete and Mixed Random Variables.

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A continuous random variable is a random variable where the data can take infinitely many values. For example, a random variable measuring the time taken for something to be done is continuous since there are an infinite number of possible times that can be taken.
For any continuous random variable with probability density function f(x), we have that:

A random variable is called discrete if it has either a finite or a countable number of possible values.

These are random variables that are neither discrete nor continuous, but are a mixture of both. In particular, a mixed random variable has a continuous part and a discrete part.

10.Write 3 properties of Gaussian random variables.

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• Gaussian random variables are completely defined through only their first and second order moments, i.e. by their means,variance, and co-variance.
• If the random variables are uncorrelated, they are also called statistically independent.
• Random variables produced by a linear transformation of X1………Xn will also be Gaussian.

1. Define Variance of Random Variable.

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The variance of random variable X is often written as Var(X) or σ2 or σ2x.

For a discrete random variable the variance is calculated by summing the product of the square of the difference between the value of the random variable and the expected value, and the associated probability of the value of the random variable, taken over all of the values of the random variable. In symbols, Var(X) = (x - µ)2 P(X = x)
An equivalent formula is, Var(X) = E(X2) – [E(X)]2
The square root of the variance is equal to the standard deviation.

2. Define Moments of a Random Variable.

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The “moments” of a random variable (or of its distribution) are expected values of powers or related functions of the random variable.

3. What is Expectation of a Random Variable?

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4. What is Markov Inequality?

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The Markov inequality applies to random variables that take only nonnegative values. It can be stated as follows:
If X is a random variable that takes only nonnegative values, then for any a>0,

5.What is Chebyshev’s Inequality?

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Chebyshev’s inequality is a probability theory that guarantees that within a specified range or distance from the mean, for a large range of probability distributions, no more than a specific fraction of values will be present. In other words, only a definite fraction of values will be found within a specific distance from the mean of a distribution.

1. What is a pair of Random Variable?

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A pair of variables whose values are determined by a random experiment is called a pair of random variables. There are two types of pairs:
• A pair of random variables is discrete if the set of values taken by each of the random variables is a finite or infinite countable set.
• A pair of random variables is continuous if the set of values taken by each of the random variables is an infinite noncountable set.

2.What is Joint PDF?

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3. What is Joint CDF?

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4. What is Joint Moments?

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Joint moments are calculated using sophisticated laboratory equipment and computer programs.
The total moment at a joint is calculated as the product of two measurable quantities:
1. the joint segments' moments of inertia, which involves knowing thee segments' masses and lengths 2. the joint's angular acceleration

5. What are the principles of software design?

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The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30.
All this is saying is that as you take more samples, especially large ones, your graph of the sample means will look more like a normal distribution.
Here’s what the Central Limit Theorem is saying, graphically. The picture below shows one of the simplest types of test: rolling a fair die. The more times you roll the die, the more likely the shape of the distribution of the means tends to look like a normal distribution graph.

6. Diffrence between Covariance and Correlation.

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1. What is Random Process?

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• A random process is a time-varying function that assigns the outcome of a random experiment to each time instant: X(t).
• For a fixed (sample path): a random process is a time varying function, e.g., a signal. -For fixed t: a random process is a random variable.
• If one scans all possible outcomes of the underlying random experiment, we shall get an ensemble of signals.
• Random Process can be continuous or discrete
• Real random process also called stochastic process – Example: Noise source (Noise can often be modeled as a Gaussian random process.

2. What is Ergodic Random Process?

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• The computation of statistical averages (e.g., mean and autocorrelation function) of a random process requires an ensemble of sample functions (data records) that may not always be feasible.
• In many real-life applications, it would be very convenient to calculate the averages from a single data record.
• This is possible in certain random processes called ergodic processes.

3. What is Wide-Sense Stationary Random Process?

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4. Define Poisson Process.

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A Poisson Process is a model for a series of discrete event where the average time between events is known, but the exact timing of events is random