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12 (10 points) Let X and Y have joint probability density function defined by f (X,Y) (x, y) = 1 2 x, if 0 < y < x < 2;• The probability of event {(X,Y)∈ B} is P(B)= X (x,y)∈B PX,Y (x,y) – Two coins, one fair, the other twoheaded A randomly chooses one and B takes the other X = ˆ 1 A gets head 0 A gets tail Y = ˆ 1 B gets head 0 B gets tail Find P(X ≥ Y) • Marginal probability mass function of X can be ob0, elsewhere Find the marginal probability density function of YYour answer should contain the region on which the marginal probability density function is nonzero 13 (10 points) Probability and CombinatoricsYou may leave your answer in terms of combinatorics symbols (n

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What is joint probability density function

What is joint probability density function-The joint probability density function (joint pdf) of X and Y is a function f(x;y) giving the probability density at (x;y) That is, the probability that (X;Y) is in a small rectangle of width dx and height dy around (x;y) is f(x;y)dxdy y d Prob = f (x;y )dxdy dy dx c x a b A joint probability density function must satisfy two properties 161 Joint density functions Recall that X is continuous if there is a function f(x) (the density) such that P(X ≤ t) = Z t −∞ fX(x)dx We generalize this to two random variables Definition 1 Two random variables X and Y are jointly continuous if there is a function fX,Y (x,y) on R2, called the joint probability density function, such

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The joint probability distribution can be expressed either in terms of a joint cumulative distribution function or in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables) These in turn can be used to find two other types of distributions the marginal distribution giving the probabilities forStack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Visit Stack ExchangeFor continuous random variables, we have the notion of the joint (probability) density function f X,Y (x,y)∆x∆y ≈ P{x < X ≤ x∆x,y < Y ≤ y ∆y} We can write this in integral form as P{(X,Y) ∈ A} = Z Z A f X,Y (x,y)dydx The basic properties of the joint density function are • f X,Y (x,y) ≥ 0 for all x and y 2

The joint probability density function of the random variables X and Y is {eq}\begin {align*} f\left ( {x,y} \right) &= \dfrac { {3x y}} {9}\;;\;1 < x < 3,\;1 < y < 2\\ & = 0Example 3513 Let X and Y be jointly continuous random variables with joint probability density function f (x, y) = ⇢ 6 e2 x e3 y for 0 < x < 1, 0 < y < 1 0 otherwise Show that X and Y are independent random variables 106How do you compute the P(x>y) for a joint density function in R?

61 Joint density functions Recall that X is continuous if there is a function f(x) (the density) such that P(X ≤ t) = Z t −∞ fX(x)dx We generalize this to two random variables Definition 1 Two random variables X and Y are jointly continuous if there is a function fX,Y (x,y) on R2, called the joint probability density function, suchIf continuous random variables \(X\) and \(Y\) are defined on the same sample space \(S\), then their joint probability density function (joint pdf) is a piecewise continuous function, denoted \(f(x,y)\), that satisfies the followingSimilar to the CDF the probability density function follows the same general rules except in two dimensions, Univariate de nition f (x) 0 for all xf (x) = d dx F(x) R 1 Since the joint density is constant then f(x;y) = c = 2 9;

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0 < y < 2 (a) erifyV that this is indeed a joint density function (b) Compute the density function of X (c) Find P ( X > Y ) (d) Find P ( YExample 3513 Let X and Y be jointly continuous random variables with joint probability density function f (x, y) = ⇢ 6 e2 x e3 y for 0 < x < 1, 0 < y < 1 0 otherwise Show that X and Y are independent random variables 106• Joint probability density function 4 REGRESSION * Line of regression The line of regression of X on Y is given by Example241 1 From the following data, find (i) The two regression equation (ii) The coefficient of correlation between the marks in Economic and Statistics (iii) The

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The joint probability density function, (,) for two continuous random variables is defined as the derivative of the joint cumulative distribution function (see Eq1) f X , Y ( x , y ) = ∂ 2 F X , Y ( x , y ) ∂ x ∂ y {\displaystyle f_{X,Y}(x,y)={\frac {\partial ^{2}F_{X,Y}(x,y)}{\partial x\partial y}}}If X is a random variable with density fx(x) and Y is a random variable with density fY(y), how would we describe the joint behavior of the tuple (X, Y) at the same time?TheUsing the replicate() function, one simulates this sampling process 1000 times, storing the outcomes in the data frame results with variable names X and YUsing the table() function, one classifies all outcomes with respect to the two variables By dividing the observed counts by the number of simulations, one obtains approximate probabilities similar to the exact probabilities shown in Table 61

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The joint probability mass function of (X;Y) is (12) p(xi;yj) = P(X = xi;Y = yj) Example 1 A fair coin is tossed three times independently let X denote the number of heads on the flrst toss and Y denote the total number of heads Find the joint probability mass function of X and Y 2The joint probability density function (joint pdf) of X and Y is a function f(x;y) giving the probability density at (x;y) That is, the probability that (X;Y) is in a small rectangle of width dx and height dy around (x;y) is f(x;y)dxdy y d Prob = f (x;y )dxdy dy dx c x a b A joint probability density function must satisfy two properties 112 (10 points) Let X and Y have joint probability density function defined by f (X,Y) (x, y) = 1 2 x, if 0 < y < x < 2;

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For continuous random variables, we have the notion of the joint (probability) density function f X,Y (x,y)∆x∆y ≈ P{x < X ≤ x∆x,y < Y ≤ y ∆y} We can write this in integral form as P{(X,Y) ∈ A} = Z Z A f X,Y (x,y)dydx The basic properties of the joint density function are • f X,Y (x,y) ≥ 0 for all x and y 2As noted in Chapter 1, the joint density function corresponds to the density of points on a scatter plot of x and y in the limit of an infinite number of points This is illustrated in Fig 33, using the data shown on the scatter plot of Fig 13 (b)If X is a random variable with density fx(x) and Y is a random variable with density fY(y), how would we describe the joint behavior of the tuple (X, Y) at the same time?The

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