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