a coin is tossed 4 times which of the following represents the probability of the coin landing on heads all 4 times

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

Find the slope of the line 25 POINTS WILL MARK BRANLIST

UNO [17]

Answer:

-7/4

Step-by-step explanation:

6 0

3 years ago

A stainless steel patio heater is a square pyramid. The length of one side of the base is 89.6 The slant height of the pyramid i

Temka [501]

Answer:

9.48

Step-by-step explanation:

Complete question

A stainless steel patio heater is a square pyramid. The length of one side of the base is 89.6 The slant height of the pyramid 90.1. What is the height of the pyramid?

To get the height of the pyramid, we will use pythagoras theorem.

slant height l is the hypotenuse

length of the base is the adjacent

height of the pyramid is the opposite

According to the theorem;

l² = h² + b²

90.1² = h² + 89.6²

8,118.01 = h²+8,028.16

h²= 8,118.01 - 8,028.16

h² = 89.85

h =√89.85

h = 9.48

Hence the height of the pyramid will be 9.48

Note that the value of the slant height was assumed to solve the problem.

3 0

3 years ago

Apply the distributive property to factor out the greatest common factor. 35 + 14 =

Galina-37 [17]

Answer: approximately 49

Step-by-step explanation:

7 0

3 years ago

Read 2 more answers

Jim wishes to buy 3 gifts that cost 15 dollars, 9 dollars, and 12 dollars. he has 1/4 of the money he needs, how much more money

Archy [21]

In order to solve this, you have to plug in all of the costs of the gifts and add them, which in all equals $36. since jim only has 1/4 of the money he needs to buy the gifts, you need to divide 36 by 4. this gives you 9, so now you know he has 9 dollars.
and to find out how much more money he needs, you can subtract 9 from 36 which represents a fourth of the required money. the answer is 27, that's how much jim needs in order to buy his gifts.

7 0

3 years ago