√100以上 p(x y) independent 102041-P(x y) independent

The events X and Y are said to be independent if the probability of X is not affected by the occurrence of Y That is, X, Y independent if and only if P(XY)=P(ot Y) Here P(XY) means theBut using the de nition of conditional probability we nd that P(Y = jX= ) = P(Y = \X= ) P(X= ) = P(Y = ) or P(Y = \X= ) = P(X= )P(Y = ) This formula is symmetric in Xand Y and so if Y is independent of Xthen Xis alsoIn real life, we usually need to deal with more than one random variable For example, if you study physical characteristics of people in a certain area, you might pick a person at random and then look at his/her weight, height, etc

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P(x y) independent

P(x y) independent-P(X) = P(Y) or P(X n Y) = 0 That is, the above is true if and only if X and Y are equally likely, or if X and Y are mutually exclusive Oh, and since we were dividing by P(X) and P(Y), both must be possible, ie nonzero probabilityDefine Z = max (X, Y), W = min (X, Y) Find the CDFs of Z and W Solution To find the CDF of Z, we can write F Z ( z) = P ( Z ≤ z) = P ( max ( X, Y) ≤ z) = P ( ( X ≤ z) and ( Y ≤ z)) = P ( X ≤ z) P ( Y ≤ z) ( since X and Y are independent) = F X ( z) F Y ( z)

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Sometimes it really is, but in general it is not Especially, Z is distributed uniformly on (1,1) and independent of the ratio Y/X, thus, P ( Z ≤ 05 Y/X) = 075 On the other hand, the inequality z ≤ 05 holds on an arc of the circle x 2 y 2 z 2 = 1, y = cx (for any given c) The length of the arc is 2/3 of the length of the circle2 Independent Random Variables The random variables X and Y are said to be independent if for any two sets of real numbers A and B, (24) P(X 2 A;Y 2 B) = P(X 2 A)P(Y 2 B) Loosely speaking, X and Y are independent if knowing the value of one of the random variables does not change the distribution of the other ran(d) YES, X and Y are independent, since fX(x)fY (y) = ˆ 2x·2y = 4xy if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 0otherwise is exactly the same as f(x,y), the joint density, for all x and y Example 4 X and Y are independent continuous random variables, each with pdf g(w) = ˆ 2w if 0 ≤ w ≤ 1 0, otherwise (a) Find P(X Y ≤ 1) (b)

Since X X X and Y Y Y are independent, P (X = a i and Y = b j) = P (X = a i) ⋅ P (Y = b j) P(X = a_i \text{ and } Y = b_j) = P(X = a_i) \cdot P(Y = b_j) P (X = a i and Y = b j ) = P (X = a i ) ⋅ P (Y = b j ) and it follows that E X ⋅ Y = ∑ i, j P (X = a i) ⋅ P (Y = b j) a i b j = (∑ i P (X = a i) a i) (∑ j P (Y = b j) b j) = E X ⋅ E YRandom variables X and Y are independent if their joint distribution function factors into the product of their marginal distribution functions • Theorem Suppose X and Y are jointly continuous random variables X and Y are independent if and only if given any two densities for X and Y their product is the joint density for the pair (X,Y) ie ProofY y' to mean the event 'X x and Y y' The joint cumulative distribution function (joint cdf) is de ned as F(x;y) = P(X x;

TEDx was created in the spirit of TED's mission, "ideas worth spreading" It supports independent organizers who want to create a TEDlike event in their own communityFigure1 f(x;y)j0 < x < 1;0 < y < 1g Note that f(x;y) is a valid pdf because P (1 < X < 1;1 < Y < 1) = P (0 < X < 1;0 < Y < 1) = Z1 1 Z1 1 f(x;y)dxdy = 6 Z1 0 Z1 0 x2ydxdy = 6 Z1 0 y 8 < Z1 0 x2dx 9 =;The notation P(xy) means P(x) given event y has occurred, this notation is used in conditional probability There are two cases if x and y are dependent or if x and y are independent Case 1) P(xy) = P(x&y)/P(y) Case 2) P(xy) = P(x)

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We say that X and Y are independent if P (X = x, Y = y) = P (X = x) P (Y = y), for all x, y In general, if two random variables are independent, then you can write P (X ∈ A, Y ∈ B) = P (X ∈ A) P (Y ∈ B), for all sets A and BFigure1 f(x;y)j0 < x < 1;0 < y < 1g Note that f(x;y) is a valid pdf because P (1 < X < 1;1 < Y < 1) = P (0 < X < 1;0 < Y < 1) = Z1 1 Z1 1 f(x;y)dxdy = 6 Z1 0 Z1 0 x2ydxdy = 6 Z1 0 y 8 < Z1 0 x2dx 9 =;Y y) Continuous case If X and Y are continuous random variables with joint density f(x;y)

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PXY=1(2) = p(2,1)/pY (1) = 01/06 = 1/6 2 If X and Y are independent Poisson RVs with respective meansY is independent of Xif P(Y = jX= ) = P(Y = ) for all ;Even if X and Y are independent, P(X ≤ x) P(Y ≤ y) ≠ P(X ≤ y) P(Y ≤ x) unless they are also identically distributed $\endgroup$ – farmer Jan 7 '19 at 2139 1

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Answer Two events, X and Y, are independent if X occurs won't impact the probability of Y occurring More examples of independent events are when a coin lands on heads after a toss and when we roll a 5 on a single 6sided die Then, when selecting a marble from a jar and the coin lands on the head after a tossLet X and Y be two discrete variables whose joint pmf has the following values p(1, 1) = 1/4 , p(1, 0) = 1/2 , p(0, 1) = 1/12 , p(0, 0) = 1/6 and is 0 elsewhere Are X and Y independ Create anIf X and Y are independent, then E(es(XY )) = E(esXesY) = E(esX)E(esY), and we conclude that the mgf of an independent sum is the product of the individual mgf's Sometimes to stress the particular rv X, we write M X(s) Then the above independence property can be concisely expressed as M

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P(y = \x= ) = p(x= )p(y = ) This formula is symmetric in Xand Y and so if Y is independent of Xthen Xis also independent of Y and we just say that Xand Y are independentDy = 6 Z1 0 y 3 dy = 1 Following the de–nition of the marginal distribution, we can get a marginal distribution for X For 0 < x < 1, f(x) ZNow assume Z independent of X given Y, and assume W independent of X and Y given Z, then we obtain P(X=x,Y=y,Z=z,W=w) = P(X=x)P(Y=yX=x)P(Z=zY=y)P(W=wZ=z) For binary variables the representation requires 1 2*1 2*1 2*1 = 1(41)*2 numbers significantly less!!

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Incoming Term: p(x y) independent,