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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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Given the following joint probability density function f_xy (x,y) = (6 x y)/8, 0Question Question 2 Let The Joint Probability Density Function For (X,Y) Be F(x,y) = I > 1, Y > 1 For R >1 And Y > 1 Find P(X > 2Y) This problem has been solved!Probability mass function p X(a)=P(X =a)=å y P X;Y(a;y) p Y(b)=P(Y =b)=å x P X;Y(x;b) In the continuous case a joint probability density function tells you the relative probability of any combination of events X =a and Y =y In the discrete case, we can define the function p X;Y nonparametrically Instead of using a formula for p we

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Probability Density Function The distribution of a random variable of continuous type is represented by the probability density function (pdf) of the random variableP(X=A) = sum P(X=A, Y=yi) for all y This is another important foundational rule in probability, referred to as the " sum rule " The marginal probability is different from the conditional probability (described next) because it considers the union of all events for the second variable rather than the probability of a single eventThe function f XY(x;y) is called the joint probability density function of Xand Y If nd Bare any sets of real numbers then by letting C= f(x;y) x2 A;y2Bgwe have P(X2A;Y 2B) = Z B Z A f XY(x;y)dxdy In words, to nd the probability that Xand Y take values in any region, we integrate the joint probability density function over that region

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For 0 x y 3 based on the area of the triangle, but we need to be careful to de ne onBivariate Distributions (Joint Probability Distributions) Sometimes certain events can be defined by the interaction of two measurements These types of events that are explained by the interaction of the two variables constitute what we call bivariate distributions When put simply, bivariate distribution means the probability that a certain event will occur when there are two independentUsing 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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See the answer Show transcribed image text Expert Answer Previous question Next question Transcribed Image Text from this Question• 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) TheFZ(z) = P(Z ≤ z) = P(X Y ≤ z) = P(Y ≤ −X z) = ˆ0 if z < 0 Rz 0 Rz−x 0 λe −λxλe−λydydx = 1−e−λz −λze−λz if z ≥ 0 This gives the pdf, fZ(z) = ˆ 0 if z < 0 λ2ze−λz if z ≥ 0, which is the pdf of a Gamma(2,λ) Thus, Z is Gamma(2,λ) random variable (b)

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P(X=A) = sum P(X=A, Y=yi) for all y This is another important foundational rule in probability, referred to as the " sum rule " The marginal probability is different from the conditional probability (described next) because it considers the union of all events for the second variable rather than the probability of a single eventFxx(x, y) = { % 0, otherwise Note e' = Find the P(X < 13 n Y < 14)The joint probability mass function of two discrete random variables $X$ and $Y$ is defined as \begin{align}%\label{} \nonumber P_{XY}(x,y)=P(X=x, Y=y) \end{align}

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Ask Question Asked 2 years, 5 months ago Active 2 years, 5 months ago Deriving the joint probability density function from a given marginal density function and conditional density function 2 Conditional Probability Mixture Model 5If 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 following f ( x, y) ≥ 0, for all ( x, y) ∈ R 2 ∬ R 2 f ( x, y) d x d y = 1 P ( ( X, Y) ∈ A) = ∬ A f ( x, y) d x d y, for any A ⊆ R 2 The first two conditions in Definition 521 provide the requirements for a function to be a valid joint pdfXY (xy) satisfies the properties of an ordinary (onevariable) density function;

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The same goes for f Y X(yx) • Probabilities via joint densities Given a region B in the xyplane, the probability that (X,Y) falls into this region is given by the double integral of f(x,y) over this region P((X,Y) ∈ B) = ZZ B f(x,y)dxdy0, 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 (n0 < x < 1 ;

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P(X=A) = sum P(X=A, Y=yi) for all y This is another important foundational rule in probability, referred to as the " sum rule " The marginal probability is different from the conditional probability (described next) because it considers the union of all events for the second variable rather than the probability of a single eventStack 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 ExchangeThe joint probability mass function of two discrete random variables $X$ and $Y$ is defined as \begin{align}%\label{} \nonumber P_{XY}(x,y)=P(X=x, Y=y) \end{align}

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Joint Probability Distributions

Joint Probability Density Function A joint probability density function for the continuous random variable X and Y, denoted as fXY(x;y), satis es the following properties 1 fXY(x;y) 0 for all x, y 2 R 1 1 R 1 1 fXY(x;y) dxdy= 1 3 For any region Rof 2D space P((X;Y) 2R) = Z Z R fXY(x;y) dxdy For when the rv's are continuous 16P ( ( X, Y) ∈ A) = ∬ A f X Y ( x, y) d x d y ( 515) The function f X Y ( x, y) is called the joint probability density function (PDF) of X and Y In the above definition, the domain of f X Y ( x, y) is the entire R 2 We may define the range of ( X, Y) as R X Y = { ( x, y) f X, Y ( x, y) > 0 }The probability that the ordered pairs of random variables (X, Y) (X,Y) (X, Y) take values in the (open or closed) intervals a, b a,b a, b and c, d, c,d, c, d, respectively, is given by the integral of a function called the joint probability density function f X Y (x, y) f_{XY} (x,y) f X Y (x, y)

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Example 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 106The function p defined for all (x i, y j) in the range space (X, Y) is called the probability function of (X, Y) The set of triplets (x i, y j;p(x i, y j)) i, j = 1, 2, is called the probability distribution of (X, Y) Joint Density Function Let (X, Y) be a continuous random variable assuming all values in some region R of the Euclidian planeIn the joint density Obtaining this probability is called marginalization, and it involves taking a weighted sum2 over the possible outcomes of the rv's that are not of interest For two variables X,Y P(X = x) = X y P(x,y) = X y P(X = xY = y)P(y) In this case P(X) is often called a marginal probability and the process of

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1 Let p(x,y) be the uniform joint probability density on the unit disk, ie, and p(x,y)=0 otherwise Calculate the pdf of XY Also find the expected value and variance of XY 2 Suppose X and Y are independent random variables, each distributed according to the exponential distribution with parameter Find the joint pdf of X and Y (easy)Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization= 10 ways Hence the probability P ( N 1 = i;N 2 = j ) = 1 10 1 i < j < 5 Q 3) (Ross # 69) The joint probability density function of X and Y is given by f ( x;y ) = 6 7 x 2 xy 2 ;

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