PTSP very short questions

PROBABILITY AND STOCASTIC PROCESSING

1. Define Probability?

A: Probability of an event is defined as

Probability of an event happening= No of ways it can happen/total no of outcomes.

2. Define a random variable?

A:  random variable is a variable whose value is unknown or a function that assigns value to each of an experiment’s outcome.

3. Define a Sample Space?

A: The sample space of an experiment is the set of all possible outcomes of that experiment.

4. Define bayes theorem?

A: The bayes theorem or bayes rule describes the probability if an event, based on prior knowledge of the conditions that might be related to the event.

5. Define probability density function?

A: The probability density function is the derivative of the probability distribution, and it id denoted by FX(x).

Fx(x) =d/dx Fx(x).

6. Define probability distribution function?

A: The probability of an event {X≤x} i.e. P{X≤x}  is known as probability distribution function and is denoted as Fx(x) and this is also called as cumulative distribution function.

FX(x) = P {X≤x} 

7. Define uniform distribution function?

A: The uniform distribution function has a random variable X restricted to a finite interval {a, b} and has f(x) as constant density over the interval. The function f(x) is defined by:

F(x) = {1/ (b-a), a≤x≤b 0, otherwise}

8. Define Gaussian random variable?

A: A random variable X is called Gaussian if its density function has the form fx(x) =1/√2πσ͓ˆ2.eˆa͓ˆ2/2σ͓ˆ2.

9. Give two properties of probability density function.

A:

·         ff X(x)X(x) > 0, for all xx ∈∈ (−∞,+∞−∞,+∞)

·          ∫+∞−∞fX(x)∫−∞+∞fX(x) = 1

10. Give two properties of probability distribution function.

A:

• If (x1<x2) = F͓(x2)≤ F͓(x1)
• P{(x1<X≤x2} = F͓(x2)-F͓(x1)

11. Define joint probability density function for two random variables.

A: Given 2 random variables X and Y, a joint probability density function or joint pdf is the density of probability for joint events i.e. if F is a subset of the real plane, then

P(F) = P(X, Y) € F =⌡⌡fx, y(y)

12. Define Marginal density function

A: These are the density functions of the single random variables X and Y and are defined as the derivative of marginal distribution function.

FX(x) =d/dx FX(x)

fy(y) =d/dyFy(y)

13. Define statistical independence.

A: The two events A and B are statistically independent if and only if

P (AnB) = P (A).P (B). Similarly X and Y are statistically independent random variables if and only if P {X≤ x, Y≤ y} = P {X≤x} P {Y≤ y}

Fxy(y)=Fx(x).Fy(y)

Fx(y)=fx(x)fy(y)

14. Defline Radom Procès.

A: A random variable, x (ζ), can be defined from a Random event, ζ, by assigning values xi to each possible outcome, Ai, of the event. Next define a Random Process, x(ζ) ,t , a function of both the event and time, by assigning to each outcome of a random event, ζ , a function in time, xi(t) , chosen from a set of functions, xi(t) .

15. Define first order stationary process.

A: A random process is called stationary to order 1 if its first order density functions with shift in time origin. i.e. f͓(x1; t1,) =FX(x1, t1+∆).

16. Define second order stationary process.

A: The process is called stationary of order 2, if its second order density function satisfies fx(x1, x2; t1, t2) = fx(x1, x2; t1+∆, t2+∆).

17. Write 3 properties of Gaussian random variables.

A:

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

18. Define Central Limit Theorem.

A: The central limit theorem states that the random variable X which is the sum of the large number of random variables always approaches the Gaussian distribution irrespective of the type of distribution each variable process and their amount of contribution into the sum.

X3=X1+X2

19. Define compound probability theorem 

A: If the probability of event A happening as a result of a trail is P (A) and after A has happened, the probability of a event B happening as a result of another trail is P (B/A), then the probability of both the events happening as a result of two trails is P (AB) or P (AnB) =P (A).P (B/A).

20. Explain deterministic and non-deterministic process.

A:

1. If the future values of any sample fn can’t be predicted exactly from the observed past values it is called non-deterministic.
2. If the future values of any sample fn can be predicted exactly from the observed past values it is called deterministic.