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Notice that the rst moment about the origin is just the mean, that is, E1X2 . 1 , we can write the variFurthermore, since the second moment about the origin is E1X 2 2 2 ance of a random variable in terms of origin moments as follows:

When an assembly is built, its current version is stored in the manifest metadata of the assembly. The version can be set by using an assembly attribute called AssemblyVersion. By default, this attribute is located in the AssemblyInfo.vb file of each project. The default value for this attribute is as follows:

The moments of a random variable can often be determined directly from the de nition in Equation S5-7, but there is an alternative procedure that is frequently useful that makes use of a special function. De nition The moment generating function of the random variable X is the expected value of e tX and is denoted by MX (t). That is, MX 1t2 E1e 2

etx f 1x2 dx,

The moment generating function MX (t) will exist only if the sum or integral in the above definition converges. If the moment generating function of a random variable does exist, it can be used to obtain all the origin moments of the random variable.

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The assembly version is made up of four numbers, as shown in Figure 15.3. The first two numbers indicate the major and minor version of the assembly. A change to the major or minor version number usually indicates that the assembly contains new functionality, either by adding or changing properties, method, or events. By default, assemblies that have different major and minor version numbers are not considered to be compatible. The third number of the version represents the revision of the assembly. The revision number is usually updated when applying service packs. No assumption of compatibility can be made between different revision numbers.

Formula Bar This appears only in Excel. Use this bar to type and edit formulas and perform calculations on your worksheet data.

Let X be a random variable with moment generating function MX (t). Then r d r MX 1t2 ` dt r t (S5-9)

x re tx f 1x2 dx,

To fix the version, manually type it:

EXAMPLE S5-5

Determine the moment generating function and use it to verify that the mean and variance of the binomial random variable are np and 2 np(1 p). From the de nition of a moment generating function, we have MX 1t2

May Be Compatible (Service Packs)

5-10

Taking the rst and second derivatives, we obtain M 1t2 X and M 1t2 X If we set t d 2MX 1t2 dt 2 npet 11 p npet 2 31 p1et 12 4 n

Always Compatible (QFEs)

12 4 n

EXAMPLE S5-6

The fourth number of the version represents the build of the assembly. Assemblies that have the same major, minor, and revision numbers, but have a different build number are considered to be compatible. An assembly with a different build number is usually deployed to correct software bugs through Quick Fix Engineering (QFE) builds. When using the asterisk to provide autonumbering of the assembly, the third number contains the number of days since January 1, 2000, and the fourth number contains the number of seconds since midnight, divided by two. Depending on the current daylight savings setting, this number may start at midnight or 1:00 A.M. The System.Version class can be used to obtain the version of an assembly. The following code has been added to the DataComponent to allow retrieval of the version of the assembly:

Find the moment generating function of the normal random variable and use it to show that the mean and variance of this random variable are and 2, respectively. The moment generating function is MX 1t2 etx 1 12 1 12 e

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