I guess you multiply everything times "x", the Xs cancel out, then you will have " a + b = 1x " basically the one does not make any change, so you will end with a " a + b = x "
I know you said "without making any assumptions," but this one is pretty important. Assuming you mean
are shape/rate parameters (as opposed to shape/scale), the PDF of
is

if
, and 0 otherwise.
The MGF of
is given by
![\displaystyle M_X(t) = \Bbb E\left[e^{tX}\right] = \int_{-\infty}^\infty e^{tx} f_X(x) \, dx = \frac{2^8}{\Gamma(8)} \int_0^\infty x^7 e^{(t-2) x} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20M_X%28t%29%20%3D%20%5CBbb%20E%5Cleft%5Be%5E%7BtX%7D%5Cright%5D%20%3D%20%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20e%5E%7Btx%7D%20f_X%28x%29%20%5C%2C%20dx%20%3D%20%5Cfrac%7B2%5E8%7D%7B%5CGamma%288%29%7D%20%5Cint_0%5E%5Cinfty%20x%5E7%20e%5E%7B%28t-2%29%20x%7D%20%5C%2C%20dx)
Note that the integral converges only when
.
Define

Integrate by parts, with


so that

Note that

By substitution, we have

and so on, down to

The integral of interest then evaluates to

so the MGF is

The first moment/expectation is given by the first derivative of
at
.
![\Bbb E[X] = M_x'(0) = \dfrac{8\times\frac12}{\left(1-\frac t2\right)^9}\bigg|_{t=0} = \boxed{4}](https://tex.z-dn.net/?f=%5CBbb%20E%5BX%5D%20%3D%20M_x%27%280%29%20%3D%20%5Cdfrac%7B8%5Ctimes%5Cfrac12%7D%7B%5Cleft%281-%5Cfrac%20t2%5Cright%29%5E9%7D%5Cbigg%7C_%7Bt%3D0%7D%20%3D%20%5Cboxed%7B4%7D)
Variance is defined by
![\Bbb V[X] = \Bbb E\left[(X - \Bbb E[X])^2\right] = \Bbb E[X^2] - \Bbb E[X]^2](https://tex.z-dn.net/?f=%5CBbb%20V%5BX%5D%20%3D%20%5CBbb%20E%5Cleft%5B%28X%20-%20%5CBbb%20E%5BX%5D%29%5E2%5Cright%5D%20%3D%20%5CBbb%20E%5BX%5E2%5D%20-%20%5CBbb%20E%5BX%5D%5E2)
The second moment is given by the second derivative of the MGF at
.
![\Bbb E[X^2] = M_x''(0) = \dfrac{8\times9\times\frac1{2^2}}{\left(1-\frac t2\right)^{10}} = 18](https://tex.z-dn.net/?f=%5CBbb%20E%5BX%5E2%5D%20%3D%20M_x%27%27%280%29%20%3D%20%5Cdfrac%7B8%5Ctimes9%5Ctimes%5Cfrac1%7B2%5E2%7D%7D%7B%5Cleft%281-%5Cfrac%20t2%5Cright%29%5E%7B10%7D%7D%20%3D%2018)
Then the variance is
![\Bbb V[X] = 18 - 4^2 = \boxed{2}](https://tex.z-dn.net/?f=%5CBbb%20V%5BX%5D%20%3D%2018%20-%204%5E2%20%3D%20%5Cboxed%7B2%7D)
Note that the power series expansion of the MGF is rather easy to find. Its Maclaurin series is

where
is the
-derivative of the MGF evaluated at
. This is also the
-th moment of
.
Recall that for
,

By differentiating both sides 7 times, we get

Then the
-th moment of
is

and we obtain the same results as before,
![\Bbb E[X] = \dfrac{(k+7)!}{7!\,2^k}\bigg|_{k=1} = 4](https://tex.z-dn.net/?f=%5CBbb%20E%5BX%5D%20%3D%20%5Cdfrac%7B%28k%2B7%29%21%7D%7B7%21%5C%2C2%5Ek%7D%5Cbigg%7C_%7Bk%3D1%7D%20%3D%204)
![\Bbb E[X^2] = \dfrac{(k+7)!}{7!\,2^k}\bigg|_{k=2} = 18](https://tex.z-dn.net/?f=%5CBbb%20E%5BX%5E2%5D%20%3D%20%5Cdfrac%7B%28k%2B7%29%21%7D%7B7%21%5C%2C2%5Ek%7D%5Cbigg%7C_%7Bk%3D2%7D%20%3D%2018)
and the same variance follows.
Answer:
So to figure that out you would take the money that he spend which was 18, each pack of gum cost 3, so to figure out the question you would simply divide 18 by 3 which would give you 6.
so how would you check that answer you could take 6 and multiple by the cost of each pack which would give you 18.
so answer is 6.
Step-by-step explanation:
18/3
=6
Answer:
To find the area of a trapezoid, take the sum of its bases, multiply the sum by ... or area_trapezoid2.gif. Where b1.gif is base1.gif , b2.gif is base2.gif , h.gif ... of a trapezoid with bases of 9 centimeters and 7 centimeters, and a height of 3 ... The area of a trapezoid is
Step-by-step explanation: