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Probability - Part 3 - Joint Probability, Bivariate Normal Distributions, Functions of Random Variable,Transformation of Random Vectors - with examples, problems and solutions

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Random Experiments with two or more Characteristics

In the first two tutorials, we have considered a random experiment that has only one characteristic and hence its outcome is a random variable X that assumes a single value. However, in the following tutorial, we will deal with random experiments having 2 (or more) characteristics and hence random variables X, Y (or more).
Such random variables are called jointly distributed rvs.

x1: height of a person

x2: weight of a person

x3: blood pressure of a person

x4: sugar count of a person

Hence, x1 , x2, x3, x4 are jointly distributed. However, here we will consider a two dimensional random variable (X,Y).

We will study the following cases:
  1. Both X and Y are discrete.
  2. Both X and Y are continuous.

Also we shall study some other characteristics of jointly distributed random variables and transformation of random vectors.

The new concepts which will be introduced to you in this tutorial

(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.

Complete Tutorial with Problems and Solutions :