# Transform methods

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The mathematical expectation E[X] of a random variable locates the center of mass for the induced distribution, and the expectation of the square of the distance between X and E[X]measures the spread of the distribution about its center of mass. These quantities are also known, respectively, as the mean (moment) of X and the variance or second moment of X about the mean. Other moments give added information. We examine the expectation of certain functions of X. Each of these functions involves a parameter, in a manner that completely determines the distribution. We refer to these as transforms. In particular, we consider three of the most useful of these: the moment generating function, the characteristic function, and the generating function for nonnegative, integer-valued random variables.

As pointed out in the units on Expectation and Variance , the mathematical expectation $E\left[X\right]={\mu }_{X}$ of a random variable X locates the center of mass for the induced distribution, and the expectation

$E\left[g\left(X\right)\right]=E\left[{\left(X-E\left[X\right]\right)}^{2}\right]=\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]={\sigma }_{X}^{2}$

measures the spread of the distribution about its center of mass. These quantities are also known, respectively, as the mean (moment) of X and the second moment of X about the mean. Other moments give added information. For example, the third moment about the mean $E\left[{\left(X-{\mu }_{X}\right)}^{3}\right]$ gives information about the skew, or asymetry, of the distribution about the mean. We investigatefurther along these lines by examining the expectation of certain functions of X . Each of these functions involves a parameter, in a manner that completely determines the distribution.For reasons noted below, we refer to these as transforms . We consider three of the most useful of these.

## Three basic transforms

We define each of three transforms, determine some key properties, and use them to study various probability distributions associated with random variables. In the section on integral transforms , we show their relationship to well known integral transforms. These have been studied extensivelyand used in many other applications, which makes it possible to utilize the considerable literature on these transforms.

Definition . The moment generating function M X for random variable X (i.e., for its distribution) is the function

${M}_{X}\left(s\right)=E\left[{e}^{sX}\right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{(}s\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{real}\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}\text{complex}\phantom{\rule{4.pt}{0ex}}\text{parameter)}$

The characteristic function φ X for random variable X is

${\phi }_{X}\left(u\right)=E\left[{e}^{iuX}\right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left({i}^{2}=-1,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}u\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{real}\phantom{\rule{4.pt}{0ex}}\text{parameter)}$

The generating function ${g}_{X}\left(s\right)$ for a nonnegative, integer-valued random variable X is

${g}_{X}\left(s\right)=E\left[{s}^{X}\right]=\sum _{k}{s}^{k}P\left(X=k\right)$

The generating function $E\left[{s}^{X}\right]$ has meaning for more general random variables, but its usefulness is greatest for nonnegative, integer-valued variables, and we limit ourconsideration to that case.

The defining expressions display similarities which show useful relationships. We note two which are particularly useful.

${M}_{X}\left(s\right)=E\left[{e}^{sX}\right]=E\left[{\left({e}^{s}\right)}^{X}\right]={g}_{X}\left({e}^{s}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\phi }_{X}\left(u\right)=E\left[{e}^{iuX}\right]={M}_{X}\left(iu\right)$

Because of the latter relationship, we ordinarily use the moment generating function insteadof the characteristic function to avoid writing the complex unit i . When desirable, we convert easily by the change of variable.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Azam
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive