Question

1. Define DeMorgan’s law.
2. Give the classical definition of Probability.
3. Define Probability using the axiomatic approach.
4. Define Moment generating function and Characteristic Function of a Random variable.

5. Define moments about origin and central moments.

6. Show that Var(kX)=k2 var(X), here k is a constant.
7. Define the statistical Independence of the Randomvariables.
8. Define point conditioning & interval conditioning distributionfunction.
9. Give the statement of central limit theorem.
10. Define correlation and covariance of two random variables X&Y.
11. Define the joint Gaussian density function of two randomvariables.

Answer 1

Answer:

1. De Morgon's Law states that the complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements. These are mentioned after the great mathematician De Morgan.

2.The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible. 

3. Probability >Axiomatic probability is a unifying probability theory. It sets down a set of axioms(rules) that apply to all of types of probability, including frequentist probability and classical probability. These rules, based on Kolmogorov's Three Axioms, set starting points for mathematicalprobability.

4. The moment generating function (MGF) of a random variable X is a function MX(s)defined as MX(s)=E[esX]. We say that MGF of X exists, if there exists a positive constant a such that MX(s) is finite for all s∈[−a,a]. Before going any further, let's look at an example.

5. Central moments.

The rth moment about themean is only defined if E[ (X - µX)r] exists. The rth momentabout the mean of a random variable X is sometimes called the rth central momentof X. The rth central momentof X about a is defined as E[ (X - a)r ]. If a = µX, we have the rth central moment of X about µX.