(For Discrete Variables X and Y) : Joint Probability, Probability mass function, Marginal Probability mass function, Conditional Probability Mass Function, Independence of events:

(For Continuous Variables X and Y) : Probability density function, Marginal Probability Distribution Function, Conditional Probability Distribution Function, Independence of events:

Properties common to both cases: Properties of CDF, Product Moments, Central moments, Non Central moments.

**A quick look at some of the examples and problems which have been solved in this tutorial.**

**Q:**Find the relation between Geometric and Pascal distributions, Exponential and Gamma distributions.

**Q:**Suppose a shopkeeper has 10 pens of a brand out of which 5 are good(G), 2 have defective inks(DI) and 3 have defective caps(DC). If 2 pens are selected at random, find the probability
i. Not more than one is DI and not more than one is DC.
ii. P(DI<2)

**Q:**The joint probability mass function of (X, Y) is given by p(x, y) = k (4x + 4y), x = 1, 2, 3; y = 0, 1, 2.
Find the (i) marginal distributions of Y
(ii)P(X ≤ 2 | Y ≤ 1).

**Q:**Check whether X and Y are independent:
P(X=1, Y=1) = 1/4, P(X=1, Y=0) = 1/4, P(X=0, Y=1) = 1/4, P(X=0, Y=0) = 1/4

**Q:**A shopping mall has parking facility for both 2-wheelers and 4-wheelers. On a randomly selected day, let X and Y be the proportion of 2 and 4 wheelers respectively. The Joint pdf of X and Y are:
f ( x, y ) = ( x + 2y ) * 2/3 ; 0 ≤ x ≤ 1; 0 ≤ y ≤ 1
=0 elsewhere
i. Find the marginal densities of X and Y.
ii. Find the probability that the proportion of two wheelers is less than half.

**Q:**Prove the additive property of: Binomial distribution, Poisson distribution.

**Q:**The amount of rainfall recorded in Jalna in June is a rv X and the amount in July is a rv Y. X and Y have a bivariate normal distribution. (X, Y) ~ (6, 4, 1, 0.25, 0.1)
Find: (i) P(X ≤ 5) (ii) P(Y ≤ 5| X = 5)

**Q:**Let X1,.....Xn be i.i.d with cdf F(x) and pdf f(x). Find the distribution of min and max of X.