Bivariate Random Variables in Java

Drawing QR Code JIS X 0510 in Java Bivariate Random Variables
Bivariate Random Variables
Decoding QR Code JIS X 0510 In Java
Using 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 = 2
Painting Denso QR Bar Code In Java
Using 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 82
Scan QR-Code In Java
Using Barcode decoder for Java Control to read, scan read, scan image in Java applications.
Table 82 X Y 1 2 f (x) 1 1/12 1/3 5/12 2 1/12 1/12 1/6 3 1/3 1/12 5/12 g(y) 1/2 1/2 1
Drawing Barcode In Java
Using Barcode encoder for Java Control to generate, create barcode image in Java applications.
Note that where the sums are over all the values of X and Y, P(X = x, Y = y) = P(Y = y) = g(y)
Bar Code Scanner In Java
Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications.
and P(X = x, Y = y) = P(X = x) = f (x)
Denso QR Bar Code Creation In Visual C#
Using Barcode generator for VS .NET Control to generate, create QR image in .NET applications.
These random variables also have expected values We nd E(X) = 1 5 1 5 +2 +3 =2 12 6 12 1 1 3 E(Y ) = 1 + 2 = 2 2 2
Encoding QR Code JIS X 0510 In .NET Framework
Using Barcode generator for ASP.NET Control to generate, create QR-Code image in ASP.NET applications.
Now we note that E(X + Y ) =
Printing Denso QR Bar Code In VS .NET
Using Barcode creation for VS .NET Control to generate, create QR Code 2d barcode image in .NET framework applications.
= E(X) + E(Y )
QR Code Creation In Visual Basic .NET
Using Barcode creator for .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications.
8
ANSI/AIM Code 128 Drawer In Java
Using Barcode printer for Java Control to generate, create Code 128C image in Java applications.
Continuous Probability Distributions
Bar Code Generation In Java
Using Barcode generator for Java Control to generate, create barcode image in Java applications.
This is not a peculiarity of this special case, but is, in fact, true for any two variables X and Y Here is a proof E(X + Y ) =
Generate Bar Code In Java
Using Barcode generation for Java Control to generate, create bar code image in Java applications.
(x + y)P(X = x, Y = y) xP(X = x, Y = y) +
RM4SCC Generator In Java
Using Barcode generation for Java Control to generate, create RM4SCC image in Java applications.
x y x y
Encoding Code 128 Code Set A In VS .NET
Using Barcode generator for .NET Control to generate, create Code128 image in .NET framework applications.
yP(X = x, Y = y) y
Print Barcode In .NET
Using 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#.NET
Using 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 .NET
Using 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#.NET
Using 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 Framework
Using 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 83
UPC-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 1
Bivariate 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 2
1 5 +2 1 2 12
+2 2 =3