Bivariate Random Variables in Java Drawing QR Code JIS X 0510 in Java Bivariate Random Variables Bivariate Random VariablesDecoding QR Code JIS X 0510 In JavaUsing Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications.Speci cally, P(Y = 1) = P(X = 1 and Y = 1) + P(X = 2 and Y = 1) + P(X = 3 and Y = 1) = 1/12 + 1/12 + 1/3 = 1/2 In a similar way, we nd P(Y = 2) = 1/2 We then found the following probability distributions for the individual variables: 5/12 if x = 1 if x = 2 f (x) = 1/6 5/12 if x = 3 and g(y) = 1/2 1/2 if y = 1 if y = 2Painting Denso QR Bar Code In JavaUsing Barcode printer for Java Control to generate, create QR Code JIS X 0510 image in Java applications.These distributions occur in the margins of the table and are called marginal distributions We have expanded Table 81 to show these marginal distributions in Table 82Scan QR-Code In JavaUsing Barcode decoder for Java Control to read, scan read, scan image in Java 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Control to generate, create Code128 image in .NET framework applications.yP(X = x, Y = y) y Print Barcode In .NETUsing Barcode generator for ASP.NET Control to generate, create barcode image in ASP.NET applications.P(X = x, Y = y) + Barcode Creator In Visual C#.NETUsing Barcode drawer for VS .NET Control to generate, create bar code image in Visual Studio .NET applications.P(X = x, Y = y)Make Data Matrix 2d Barcode In .NETUsing Barcode encoder for ASP.NET Control to generate, create Data Matrix image in ASP.NET applications.xf (x) + Encode ANSI/AIM Code 39 In C#.NETUsing Barcode maker for .NET framework Control to generate, create Code 39 Full ASCII image in VS .NET applications.y g(y) = E(X) + E(Y )EAN-13 Generation In .NET FrameworkUsing Barcode maker for .NET framework Control to generate, create EAN-13 image in .NET framework applications.This is easily extended to any number of random variables: E(X + Y + Z + ) = E(X) + E(Y ) + E(Z) + When more than one random variable is de ned on the same sample space, they may be related in several ways: they may be totally dependent as, for example, if X = Y or if X = Y 4, they may be totally independent of each other, or they may be partially dependent on each other In the latter case, the variables are called correlated This will be dealt with when we consider the subject of regression later It is important to emphasize that E(X + Y + Z + ) = E(X) + E(Y ) + E(Z) + no matter what the relationships are between the several variables, since no condition was used in the proof above Note in the example we have been considering that P(X = 1 and Y = 1) = 1/12 = P(X = 1) P(Y = 1) = 5/12 1/2 = 5/24 / so X and Y are not independent Now consider an example where X and Y are independent of each other We show another joint probability distribution function in Table 83UPC-A Supplement 5 Generation In Visual C#Using Barcode maker for VS .NET Control to generate, create UPC-A Supplement 2 image in .NET framework applications.Table 83 X y 1 2 f (x) 1 5/24 5/24 5/12 2 1/12 1/12 1/6 3 5/24 5/24 5/12 g(y) 1/2 1/2 1Bivariate Random Variables Note that P(X = x, Y = y) = P(X = x)P(Y = y) in each case, so the random variables are independent We can calculate, for example, P(X = 1 and Y = 1) = 5/24 = P(X = 1) P(Y = 1) = 1/12 1/2 The other entries in the table can be checked similarly Here we have shown the marginal distributions of the random variables X and Y , f (x) and g(y), in the margins of the table Now consider the random variable X Y and in particular its expected value Using the fact that X and Y are independent, we sum the values of X Y multiplied by their probabilities to nd the expected value of the product of X and Y : E(X Y ) = 1 1 1 2 5 1 1 5 +1 2 +1 3 12 6 2 12 1 1 5 1 +2 3 6 2 12 21 5 +2 1 2 12+2 2 =3