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2 years ago

State if each triangle is a right triangle.

Mathematics

1 answer:

Kitty [74]2 years ago

3 0

Answer:

No, It is not a right triangle

Step-by-step explanation:

For it to be a right triangle, the hypotenuse would be equal to the sum of the legs when squared.

Use the pythagorean theorem-

8^2+12^2=15^2

64+144=225

208=225

Since they aren't equal, it is not a right triangle.

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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 $\lambda=1$ .

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2 years ago

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nikklg [1K]

3.6.2x: x/36 or you can use cymath

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2 years ago

1. Noah had $ 5.25 and Lin had $ 8.95. How much did they have altogether?

amid [387]

1. 5.25 + 8.95 = 14.20

2. 21.32 - 9.50 = 11.82

3. 24.50 × 5 = 122.5

4. 85.20 ÷ 4 = 21.3

Hope this helped! Please comment if you have any questions regarding how I got my answers. Stay safe.

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2 years ago

A 34-foot tall utility pole is supported by two wires that are anchored 8 feet from the

Andrew [12]

Answer:34.93

Step-by-step explanation:

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2 years ago

joshua drinks 8 cups of water a day. The recommended daily amout is given in fluid ounces.How many fluid ounces of water does he

olga nikolaevna [1]

1 cup = 8 fluid oz

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So Joshua drinks 64 fluid ounces a day

Hope this helped. Have a great day!

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2 years ago

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