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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

After reading this tutorial you might want to check out some of our other Mathematics Quizzes as well.
 Quizzes on ProgressionsMCQ #1: Arithmetic Progression MCQ #2: Geometric ProgressionMCQ #3 : More on Geometric Progressions.MCQ #4 : Harmonic Progressions. MCQ #5: More on Harmonic ProgressionMCQ #6: Mixed ProgressionsComplex NumbersMCQ #1MCQ #2: More on Complex NumbersQuadratic EquationsMCQ Quadratic EquationsQuadratic In-equationsMCQ Quadratic In-equations Coordinate Geometry - Straight LinesMCQ #1: Cartesian Planes, Straight Line BasicsMCQ #2 on Straight LinesMCQ #3 on Straight LinesMCQ #4 on Straight LinesCircles1 MCQ #1 on Circles. 2 MCQ #2 on Circles. 3 MCQ #3 on Circles. Conic Sections- Parabola, Hyperbola, Ellipse1 MCQ- The Basics of Conic Sections2 MCQ on Parabola..3 MCQ on Hyperbola4 MCQ on Ellipses. ProbabilityMCQ #1 on Basic ProbabilityMCQ #2: More Challenging Problems on ProbabilityMCQ #3- Conditional Probability and Bayes Theorem

## 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.
Ex:

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:

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 :

 Probability - Part Zero - A Very Basic Introduction Probability - Part 1 - Basic Probability Definitions, Random Variables Probability - Part 2 - A Tutorial on Probability Distributions Probability - Part 3 - Joint Probability, Bivariate Normal Distributions, Functions of Random Variable,Transformation of Random Vectors