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Jointly distributed Random variables

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## Jointly distributed Random variables

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**Jointly distributed Random variables**Multivariate distributions**Quite often there will be 2 or more random variables (X, Y,**etc) defined for the same random experiment. Example: A bridge hand (13 cards) is selected from a deck of 52 cards. X = the number of spades in the hand. Y = the number of hearts in the hand. In this example we will define: p(x,y) = P[X = x, Y = y]**The function**p(x,y) = P[X = x, Y = y] is called the joint probability function of X and Y.**Note:**The possible values of X are 0, 1, 2, …, 13 The possible values of Y are also 0, 1, 2, …, 13 and X + Y ≤ 13. The number of ways of choosing the yhearts for the hand The number of ways of choosing the xspades for the hand The number of ways of completing the hand with diamonds and clubs. The total number of ways of choosing the 13 cards for the hand**Bar graph: p(x,y)**p(x,y) y x**Example:**A die is rolled n = 5 times X = the number of times a “six”appears. Y = the number of times a “five”appears. Now p(x,y) = P[X = x, Y = y] The possible values of X are 0, 1, 2, 3, 4, 5. The possible values of Y are 0, 1, 2, 3, 4, 5. and X + Y ≤ 5**A typical outcome of rolling a die n = 5 times will be a**sequence F5FF6 where F denotes the outcome {1,2,3,4}. The probability of any such sequence will be: where x = the number of sixes in the sequence and y = the number of fives in the sequence**Now**p(x,y) = P[X = x, Y = y] Where K = the number of sequences of length 5 containing x sixes and y fives.**Thus**p(x,y) = P[X = x, Y = y] if x + y ≤ 5.**Bar graph: p(x,y)**p(x,y) y x**General properties of the joint probability function;**p(x,y) = P[X = x, Y = y]**Example:**A die is rolled n = 5 times X = the number of times a “six”appears. Y = the number of times a “five”appears. What is the probability that we roll more sixes than fives i.e. what is P[X > Y]?**Definition:**Let X and Y denote two discrete random variables with joint probability function p(x,y) = P[X = x, Y = y] Then pX(x) = P[X = x] is called the marginal probability function of X. and pY(y) = P[Y = y] is called the marginal probability function of Y.**Note: Let y1, y2, y3, … denote the possible values of Y.**Thus the marginal probability function of X, pX(x) is obtained from the joint probability function of X and Y by summing p(x,y) over the possible values of Y.**Example:**A die is rolled n = 5 times X = the number of times a “six”appears. Y = the number of times a “five”appears.**Definition:**Let X and Y denote two discrete random variables with joint probability function p(x,y) = P[X = x, Y = y] Then pX |Y(x|y) = P[X = x|Y = y] is called the conditional probability function of X given Y = y and pY |X(y|x) = P[Y = y|X = x] is called the conditional probability function of Y given X = x**Note**and**Marginal distributions describe how one variable behaves**ignoring the other variable. • Conditional distributions describe how one variable behaves when the other variable is held fixed**Example:**A die is rolled n = 5 times X = the number of times a “six”appears. Y = the number of times a “five”appears. y x**The conditional distribution of X given Y = y.**pX |Y(x|y) = P[X = x|Y = y] y x**The conditional distribution of Y given X = x.**pY |X(y|x) = P[Y = y|X = x] y x**Example**A Bernoulli trial (S - p, F – q = 1 – p) is repeated until two successes have occurred. X = trial on which the first success occurs and Y = trial on which the 2nd success occurs. Find the joint probability function of X, Y. Find the marginal probability function of X and Y. Find the conditional probability functions of Y given X = x and X given Y = y,**Solution**x y A typical outcome would be: FFF…FSFFF…FS x - 1 y – x - 1**The marginal distribution of X**This is the geometric distribution**The marginal distribution of Y**This is the negative binomial distribution with k = 2.**The conditional distribution of X given Y = y**This is the geometric distribution with time starting at x.**The conditional distribution of Y given X = x**This is the uniform distribution on the values 1, 2, …(y – 1)**The joint probability function;**p(x,y) = P[X = x, Y = y]**Definition: Two random variable are said to have joint**probability density function f(x,y) if**If**then defines a surface over the x – y plane**A**f(x,y) A**y**d c x a b If the region A = {(x,y)| a ≤ x ≤ b, c ≤ y ≤ d} is a rectangular region with sides parallel to the coordinate axes: Then A f(x,y)**To evaluate**A First evaluate the inner integral Then evaluate the outer integral f(x,y)**y**d c x a b dy y = area under surface above the line where y is constant Infinitesimal volume under surface above the line where y is constant f(x,y)**The same quantity can be calculated by integrating first**with respect to y, than x. A First evaluate the inner integral Then evaluate the outer integral f(x,y)**y**d c x a b dx x = area under surface above the line where x is constant Infinitesimal volume under surface above the line where x is constant f(x,y)**Example: Compute**Now f(x,y)**The same quantity can be computed by reversing the order of**integration f(x,y)**y**Suppose the region A is defined as follows A = {(x,y)| a(y)≤ x ≤ b(y),c ≤ y ≤ d} d c x Then b(y) a(y) A