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grigory [225]
3 years ago
12

What is the function rule for (1,6) (2,24) and (3,54)?

Mathematics
1 answer:
NeX [460]3 years ago
7 0
First, find a pattern in the relationship between the two numbers in each set:

1*6 = 6, 1 + 5 = 6, 1^n*6 = 6

2*12 = 24, 2 + 22 = 24, 2^2*6, 2^3*3 = 24

3*18 = 54, 3 + 52 = 54, 3^2*6 = 54, 3^3*2 = 54

Sorting through the possible relationships reveals that in each case the first value in (x,y) relates to the second by the equation x^2*6 = y
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If X is a r.v. such that E(X^n)=n! Find the m.g.f. of X,Mx(t). Also find the ch.f. of X,and from this deduce the distribution of
astraxan [27]
M_X(t)=\mathbb E(e^{Xt})
M_X(t)=\mathbb E\left(1+Xt+\dfrac{t^2}{2!}X^2+\dfrac{t^3}{3!}X^3+\cdots\right)
M_X(t)=\mathbb E(1)+t\mathbb E(X)+\dfrac{t^2}{2!}\mathbb E(X^2)+\dfrac{t^3}{3!}\mathbb E(X^3)+\cdots
M_X(t)=1+t+t^2+t^3+\cdots
M_X(t)=\displaystyle\sum_{k\ge0}t^k=\frac1{1-t}

provided that |t|.

Similarly,

\varphi_X(t)=\mathbb E(e^{iXt})
\varphi_X(t)=1+it+(it)^2+(it)^3+\cdots
\varphi_X(t)=(1-t^2+t^4-t^6+\cdots)+it(1-t^2+t^4-t^6+\cdots)
\varphi_X(t)=(1+it)(1-t^2+t^4-t^6+\cdots)
\varphi_X(t)=\dfrac{1+it}{1+t^2}=\dfrac1{1-it}

You can find the CDF/PDF using any of the various inversion formulas. One way would be to compute

F_X(x)=\displaystyle\frac12+\frac1{2\pi}\int_0^\infty\frac{e^{itx}\varphi_X(-t)-e^{-itx}\varphi_X(t)}{it}\,\mathrm dt

The integral can be rewritten as

\displaystyle\int_0^\infty\frac{2i\sin(tx)-2it\cos(tx)}{it(1+t^2)}\,\mathrm dt

so that

F_X(x)=\displaystyle\frac12+\frac1{2\pi}\int_0^\infty\frac{\sin(tx)-t\cos(tx)}{t(1+t^2)}\,\mathrm dt

There are lots of ways to compute this integral. For instance, you can take the Laplace transform with respect to x, which gives

\displaystyle\mathcal L_s\left\{\int_0^\infty\frac{\sin(tx)-t\cos(tx)}{t(1+t^2)}\,\mathrm dt\right\}=\int_0^\infty\frac{1-s}{(1+t^2)(s^2+t^2)}\,\mathrm dt
=\displaystyle\frac{\pi(1-s)}{2s(1+s)}

and taking the inverse transform returns

F_X(x)=\dfrac12+\dfrac1\pi\left(\dfrac\pi2-\pi e^{-x}\right)=1-e^{-x}

which describes an exponential distribution with parameter \lambda=1.
6 0
3 years ago
Which is more, 7,037 yards or 4 miles
balandron [24]
4 miles are about 7040 yards so they are very close but a mile is longer.
8 0
2 years ago
Please please please help fast this is a BIG deal
aniked [119]

Answer:

190.625 ft^2

Step-by-step explanation:

A = bh = (15.25) * (12.5) = 190.625 ft^2

5 0
2 years ago
Read 2 more answers
(NEED DONE SOON) The measure of two supplementary angles are 4x - 24 and 4(2x - 3). <br> What is x
Zarrin [17]

Answer:

x = 12

Step-by-step explanation:

A pair of supplementary angles added together MUST equal 180. Therefore, as you are given two angles with the measures of (4x - 24) and 4(2x - 3), they must add together to equal 180. So, we get this equation:

(4x - 24) + 4(2x - 3) = 180

4x - 24 + 8x - 12 = 180 (we distributed 4(2x-3))

12x - 36 = 180

12x = 144

x = 12, -12

Therefore, x is 12 because -12 would mean that (4x-24) and 4(2x-3) are both negative, which is impossible for an angle measure.

5 0
2 years ago
What is the second term of (p+q)^5<br> 1. 25p^4<br> 2. 5p^4q<br> 3. 25p^4q<br> 4. 125p^4q
yaroslaw [1]
I hope this helps you

4 0
2 years ago
Read 2 more answers
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