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3 years ago

Which statement must be true to prove | || m?

Mathematics

1 answer:

Yakvenalex [24]3 years ago

6 0

Answer C is correct. If angle 1 + angle 2 = 180 degrees then L is parallel to M.
Co interior angles are only supplementary in parallel lines

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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

Measured in meters the wavelength of red light is about 8 times 10^-7, and the wave Leung violet light is 4 times 10^-7. The wav

zubka84 [21]

Answer:

C. 2 times as long

Step-by-step explanation

8 times 10^-7 written in standard form is 0.0000008

4 times 10^-7 written in standard form is 0.0000004

divide both of them:

4 0

3 years ago

Read 2 more answers

Maya designed two intersecting roads. She drew the roads so that one of the angles at the intersection was 45 degrees. What are

Citrus2011 [14]

The sum of opposite angles are equal, so two of the angles are 45°. The sum of all angles about the intersection of two lines is 360°. So the remaining two angles are found by:

α=(360-2*45)/2

α=135° thus all four angles are:

45°,135°,45°,135°

4 0

3 years ago

If I decline by 23%, what percent would be remaining?

egoroff_w [7]

Answer: 77%

Step-by-step explanation:

100-23=77

Hope it helps!

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3 years ago

What would the inter-rater reliability be for a 50-item measure in which the number of agreements between Rater 1 and Rater 2 wa

nlexa [21]

What would the inter-rater reliability be for a 50-item measure in which the number of agreements between Rater 1 and Rater 2 was 45?
C) 0.90

7 0

3 years ago