Introduction for generating function method:
In generating function method, a moment generating function is a real valued function from which one can generate all the moments of a given random variable. In many cases, it is easier to compute various moments of X using the moment generating function.
In other words, the moment generation function may be considered an expression containing all the moment of a probability distribution function, p(|x).
Definition for Generating Function Method:
In generating function method, Let X be a random variable whose probability density function is f(x). A real valued function M : `RR` `->` `RR` defined by
M(t) = E(`e^(tX)`)
is called the moment generating function of X if this expected value exists for all t in the interval −h < t < h for some h > 0.
In general, not every random variable has a moment generating function. But if the moment generating function of a random variable exists, then it is unique. At the end of this section, we will give an example of a random variable, which does not have a moment generating function or generating function method,
Using the definition of expected value of a random variable, we obtain the explicit representation for M(t) as
`M_x`(t) `=` E(`e^(tX)`) `=` `sum_x` `e^(tx)` f (x)
if X is a discrete random variable and
`M_x(t)` `=` E(`e^(tX)`) `=`int_(-oo)^(oo)` `e^(tx)` f(x) dx
if X is a continuous random variable.Having problem with free homework help math keep reading my upcoming posts, i will try to help you.
Example for Generating Function Method:
In generating function method, let X is a random variable whose moment generating function is M(t) and n be any natural number. What is the nth derivative of M(t) at t = 0?
Solution:
`d/(dt)`M(t) `=` `d/(dt)` E(`e^(tX)`)
`=` E(`d/(dt)` `xx` `e^(tX)`)
`=` E(`X` `e^(tX)`).
Similarly,
`((d)^(2))/(d(t)^(2))`M(t) `=` `((d)^(2))/(d(t)^(2))` E(`e^(tX)`)
`=` E(`((d)^(2))/(d(t)^(2))` `xx` `e^(tX)`)
`=` E(`X^2` `e^(tX)`).
Hence, in general we get
`((d)^(n))/(d(t)^(n))`M(t) `=` `((d)^(2))/(d(t)^(2))` E(`e^(tX)`)
`=` E(`((d)^(n))/(d(t)^(n))` `xx` `e^(tX)`)
`=` E(`X^n` `e^(tX)`).
If we set t`=` 0 in the `n^("th")` derivative, we get
`((d)^(n))/(d(t)^(n))`M(t) `|_(t=0)` `=` E(`X^n` `e^(tX)` `|_(t=0)` `=` E(`X^n`).
Hence the `n^("th")` derivative of the moment generating function of X evaluated at t= 0 is the `n^("th")` moment of X about the origin.
In generating function method, this example tells us if we know the moment generating function of a random variable; then we can generate all the moments of X by taking derivatives of the moment generating function and then evaluating them at zero.
In generating function method, a moment generating function is a real valued function from which one can generate all the moments of a given random variable. In many cases, it is easier to compute various moments of X using the moment generating function.
In other words, the moment generation function may be considered an expression containing all the moment of a probability distribution function, p(|x).
Definition for Generating Function Method:
In generating function method, Let X be a random variable whose probability density function is f(x). A real valued function M : `RR` `->` `RR` defined by
M(t) = E(`e^(tX)`)
is called the moment generating function of X if this expected value exists for all t in the interval −h < t < h for some h > 0.
In general, not every random variable has a moment generating function. But if the moment generating function of a random variable exists, then it is unique. At the end of this section, we will give an example of a random variable, which does not have a moment generating function or generating function method,
Using the definition of expected value of a random variable, we obtain the explicit representation for M(t) as
`M_x`(t) `=` E(`e^(tX)`) `=` `sum_x` `e^(tx)` f (x)
if X is a discrete random variable and
`M_x(t)` `=` E(`e^(tX)`) `=`int_(-oo)^(oo)` `e^(tx)` f(x) dx
if X is a continuous random variable.Having problem with free homework help math keep reading my upcoming posts, i will try to help you.
Example for Generating Function Method:
In generating function method, let X is a random variable whose moment generating function is M(t) and n be any natural number. What is the nth derivative of M(t) at t = 0?
Solution:
`d/(dt)`M(t) `=` `d/(dt)` E(`e^(tX)`)
`=` E(`d/(dt)` `xx` `e^(tX)`)
`=` E(`X` `e^(tX)`).
Similarly,
`((d)^(2))/(d(t)^(2))`M(t) `=` `((d)^(2))/(d(t)^(2))` E(`e^(tX)`)
`=` E(`((d)^(2))/(d(t)^(2))` `xx` `e^(tX)`)
`=` E(`X^2` `e^(tX)`).
Hence, in general we get
`((d)^(n))/(d(t)^(n))`M(t) `=` `((d)^(2))/(d(t)^(2))` E(`e^(tX)`)
`=` E(`((d)^(n))/(d(t)^(n))` `xx` `e^(tX)`)
`=` E(`X^n` `e^(tX)`).
If we set t`=` 0 in the `n^("th")` derivative, we get
`((d)^(n))/(d(t)^(n))`M(t) `|_(t=0)` `=` E(`X^n` `e^(tX)` `|_(t=0)` `=` E(`X^n`).
Hence the `n^("th")` derivative of the moment generating function of X evaluated at t= 0 is the `n^("th")` moment of X about the origin.
In generating function method, this example tells us if we know the moment generating function of a random variable; then we can generate all the moments of X by taking derivatives of the moment generating function and then evaluating them at zero.
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