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Section 51 Joint Distributions of Continuous RVs Example 1, cont Based on the CDF we can calculate the pdf using the 2nd partial derivative with regard to x and yThe weight of each bottle (Y) and the volume of laundry detergent it contains (X) are measured Marginal probability distribution If more than one random variable is defined in a random experiment, it is important to distinguish between the joint probability distribution of X and Y and the probability distribution of each variable individuallyTrue, then automatically the statement ∀x ∈ A,∃y ∈ B,P(x,y) must be true (but in general it doesn't go the other way) Aside Occasionally, you will see a nested quantifier at the end of a statement, in which case it is implied that the quantifier is the last in terms of order For example, here is the definition of bounded

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What does p(x y) mean-Resolution Example and Exercises Solutions to Selected Problems Example Consider the following axioms All hounds howl at night Anyone who has any cats will not have any mice_____ Example Let U = {1,2,3} Find an expression equivalent to ∀ x∃ yP (x,y) where the variables are bound by substitution instead Discrete Mathematics by Section 13 and Its Applications 4/E Kenneth Rosen TP 7 Expand from inside out or outside in

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Suppose P(x,y) is the predicate x/y=1?P(X = 1) = 3/8 ;P(x;y) p Y (y) = P a

Making a Formula Now imagine we want the chances of 5 heads in 9 tosses to list all 512 outcomes will take a long time!F (x, y) = ax^2 by is a joint probability distribution function of X, Y and 0 less than x less than 1, 0 less than y less than 2 Given that a b = 2, Find P(x less than 05) Determine theExamples x, y (universe of discourse can be people, students, numbers) x y P(x,y) 9 M Hauskrecht Nested quantifiers • More than one quantifier may be necessary to capture the meaning of a statement in the predicate logic Example • There is a person who loves everybody

P(X = 3) = 1/8 ;P(X = 1) = 3/8 ;And this is what it looks like as a graph It is symmetrical!

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And this is what it looks like as a graph It is symmetrical!The example in the introduction demonstrated events that were clearly independent However, it can sometimes be a challenge to identify whether events are independent or not Consider the following example Y = b) = P(X = a) \cdot P(Y = b) P (X = a andMaking a Formula Now imagine we want the chances of 5 heads in 9 tosses to list all 512 outcomes will take a long time!

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• Example X = {x, y, z}, and P = {(x, y), (x, z), (y, z)}, ie x < p y, x < p z, y < p z • One utility representation is u (x) = 5, u (y) = 3, u (z) = 1 • Another utility representation is u (x) = 2 3, u (y) = 0, u (z) =17 • In general, if u represents P, then so does f u, for any strictly increasing function f R → R Dana Foarta GThe Bernoulli differential equation is an equation of the form y ′ p (x) y = q (x) y n y' p(x) y=q(x) y^n y ′ p (x) y = q (x) y nThis is a nonlinear differential equation that can be reduced to a linear one by a clever substitution The new equation is a first order linear differential equation, and can be solved explicitlyThe Bernoulli equation was one of the first differentialSo let's make a formula In our previous example, how can we get the values 1, 3, 3 and 1 ?

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Examples x, y (universe of discourse can be people, students, numbers) x y P(x,y) 9 M Hauskrecht Nested quantifiers • More than one quantifier may be necessary to capture the meaning of a statement in the predicate logic Example • There is a person who loves everybodyP x,y f X,Y (x,y) = 1 The distribution of an individual random variable is call the marginal distribution The marginal mass function for X is found by summing over the appropriate column and the marginal mass functionNamely, everywhere that the original formula has an " x ", I will now plug in an " x h "

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The probability mass function, f(x) = P(X = x), of a discrete random variable X has the following properties All probabilities are positive fx(x) ≥ 0 Any event in the distribution (eg "scoring between and 30") has a probability of happening of between 0 and 1 (eg 0% and 100%)P(X = 1) = 3/8 ;Find the probability a student's ankle diameter measurement is at 50 percent greater than her wrist diameter measurement, that is \(P(Y > 15 X)\) Simulating Bivariate Normal measurements The computation of the probability \(P(Y > 15 X)\) is not obvious from the information provided But simulation provides an attractive method of

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Uniqueness Quantifier 9!x P(x) means that there existsone and only one x in the domain such that P(x) is true 91x P(x) is an alternative notation for 9!x P(x) This is read as I There is one and only one x such that P(x) I There exists a unique x such that P(x) Example Let P(x) denote x 1 = 0 and U are the integers Then 9!x P(x) is trueP(X = 3) = 1/8 ;Suppose P(x,y) is the predicate x/y=1?

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2D Geometrical Transformations Assumption Objects consist of points and lines A point is represented by its Cartesian coordinates P = (x, y)Geometrical Transformation Let (A, B) be a straight line segment between the points A and BSection 51 Joint Distributions of Continuous RVs Example 1, cont Based on the CDF we can calculate the pdf using the 2nd partial derivative with regard to x and yP(X = 0) = 1/8 ;

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P(X = 2) = 3/8 ;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 2Example \(\PageIndex{1}\) For an example of conditional distributions for discrete random variables, we return to the context of Example 511, where the underlying probability experiment was to flip a fair coin three times, and the random variable \(X\) denoted the number of heads obtained and the random variable \(Y\) denoted the winnings when betting on the placement of the first heads

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1 2 3 f (x,y) x 1 2 025 0 025 025 0 025 025 05 025 05 05 f Y (y) f X (x) so t hat μ X = 3 / 2, μ Y = 2, σ X = 1 / 2, and σ Y = √ 1 / 2 1 What is the covariance of \(X\) and \(Y\)?Note that conditions #1 and #2 in Definition 511 are required for \(p(x,y)\) to be a valid joint pmf, while the third condition tells us how to use the joint pmf to find probabilities for the pair of random variables \((X,Y)\)Stack 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 Exchange

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Note that as usual, the comma means "and," so we can write \begin{align}%\label{} \nonumber P_{XY}(x,y)&=P(X=x, Y=y) \\ \nonumber &= P\big((X=x)\textrm{ and }(Y=y)\big)Example 5 X and Y are jointly continuous with joint pdf f(x,y) = (e−(xy) if 0 ≤ x, 0 ≤ y 0, otherwise Let Z = X/Y Find the pdf of Z The first thing we do is draw a picture of the support set (which in this case is the first3 (pX,Y) is the coordinates of a randomly selected point from the disk {(x,y) x2 y2 ≤ 2} Find the joint density of (X,Y) Calcualte P(X < Y) and the probability that (X,Y) is in the unit disk {(x,y) p x2 y2 ≤ 1}

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Continuous Random Variables can be either Discrete or Continuous Discrete Data can only take certain values (such as 1,2,3,4,5) Continuous Data can take any value within a range (such as a person's height)Prolog examples with explanations Prolog always performs depthfirstsearch, Matches facts & rules (ie knowledge base) in topdown manner and resolves the goals or subgoals in lefttoright manner Most important thing to keep in mind while writing prolog program "order of writing facts & rules always matters"X is a value that X can take;

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The subject as an argument (to the functional symbol) P (x) Examples Father( x) unary predicate Brother( x,y) binary predicate Sum( x,y,z) ternary predicate P( x,y,z,t) nary predicate 3/1 Predicate Logic and Quanti ers CSE235 Propositional Functions De nition A statement of the form P (x1;x2;;xn) is the value of the propositionalP(x;y) p Y (y) = P aFind the probability a student's ankle diameter measurement is at 50 percent greater than her wrist diameter measurement, that is \(P(Y > 15 X)\) Simulating Bivariate Normal measurements The computation of the probability \(P(Y > 15 X)\) is not obvious from the information provided But simulation provides an attractive method of

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_____ Example Let U = {1,2,3} Find an expression equivalent to ∀ x∃ yP (x,y) where the variables are bound by substitution instead Discrete Mathematics by Section 13 and Its Applications 4/E Kenneth Rosen TP 7 Expand from inside out or outside inGiven that f (x) = 3x 2 2x, find f (x h) This one feels wrong, because it's asking me to plug something that involves x in for the original x But this evaluation works exactly like all the others;Notice the different uses of X and x X is the Random Variable "The sum of the scores on the two dice";

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We say that X is a subset of Y, since every element of X is also in Y This is denoted by A Venn diagram for the relationship between these sets is shown to the right Answer X is a subset of Y Example 3 Given P = {1, 3, 4} and QThe example in the introduction demonstrated events that were clearly independent However, it can sometimes be a challenge to identify whether events are independent or not Consider the following example Y = b) = P(X = a) \cdot P(Y = b) P (X = a andP(X = 0) = 1/8 ;

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P(X = 0) = 1/8 ;Let's say there are two different expressions, P(x, y), and P(a, f(z)) In this example, we need to make both above statements identical to each other For this, we will perform the substitution2D Geometrical Transformations Assumption Objects consist of points and lines A point is represented by its Cartesian coordinates P = (x, y)Geometrical Transformation Let (A, B) be a straight line segment between the points A and B

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Making a Formula Now imagine we want the chances of 5 heads in 9 tosses to list all 512 outcomes will take a long time!3 (pX,Y) is the coordinates of a randomly selected point from the disk {(x,y) x2 y2 ≤ 2} Find the joint density of (X,Y) Calcualte P(X < Y) and the probability that (X,Y) is in the unit disk {(x,y) p x2 y2 ≤ 1}P x,y f X,Y (x,y) = 1 The distribution of an individual random variable is call the marginal distribution The marginal mass function for X is found by summing over the appropriate column and the marginal mass function

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The order of quantifiers (example) Assume P(x,y) is (xy = 6) Translate the following statement into English x y P(x,y) domain integers Solution There is an integer x for which there is an integer y that xy = 6 There is a pair of integers x, y for which xy = 6 8Tion fXY(x;y) = P(X = x;Y = y) For example, we have fXY(129;15) = 012 5 If we are given a joint probability distribution for Xand Y, we can obtain the individual probability distribution for Xor for Y (and these are called the Marginal Probability DisSo let's make a formula In our previous example, how can we get the values 1, 3, 3 and 1 ?

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Prolog examples with explanations Prolog always performs depthfirstsearch, Matches facts & rules (ie knowledge base) in topdown manner and resolves the goals or subgoals in lefttoright manner Most important thing to keep in mind while writing prolog program "order of writing facts & rules always matters"And this is what it looks like as a graph It is symmetrical!P(X = 2) = 3/8 ;

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P(X = 2) = 3/8 ;Example \(\PageIndex{1}\) For an example of conditional distributions for discrete random variables, we return to the context of Example 511, where the underlying probability experiment was to flip a fair coin three times, and the random variable \(X\) denoted the number of heads obtained and the random variable \(Y\) denoted the winnings when betting on the placement of the first heads_____ Example Let U = {1,2,3} Find an expression equivalent to ∀ x∃ yP (x,y) where the variables are bound by substitution instead Discrete Mathematics by Section 13 and Its Applications 4/E Kenneth Rosen TP 7 Expand from inside out or outside in

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P(X = 3) = 1/8 ;Suppose P(x,y) is the predicate x/y=1?We say that X is a subset of Y, since every element of X is also in Y This is denoted by A Venn diagram for the relationship between these sets is shown to the right Answer X is a subset of Y Example 3 Given P = {1, 3, 4} and Q

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2D Geometrical Transformations Assumption Objects consist of points and lines A point is represented by its Cartesian coordinates P = (x, y)Geometrical Transformation Let (A, B) be a straight line segment between the points A and B\(f_Y(y)=\sum\limits_x f(x,y)=P(Y=y),\qquad y\in S_2\) where, for each \(y\) in the support \(S_2\), the summation is taken over all possible values of \(x\) If you again take a look back at the representation of our joint pmf in tabular form, you might notice that the following holds true2D Geometrical Transformations Assumption Objects consist of points and lines A point is represented by its Cartesian coordinates P = (x, y)Geometrical Transformation Let (A, B) be a straight line segment between the points A and B

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Again ∀x ∀y P(x, y) is equivalent to ∀y ∀x P(x, y) However, when the nested quantifiers are not same, changing the order changes meaning of statement Example4 Assume P(x, y, z) is (x y = z) ∀x ∀y ∃z P(x, y, z) domain real numbers Translates toFor all real numbers x and y there is a real number z such that x y = z (True)So let's make a formula In our previous example, how can we get the values 1, 3, 3 and 1 ?

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