What's the value of of x

Show that if X is a geometric random variable with parameter p, then

Answer:

$\sum_{k=1}^{\infty} \frac{p(1-p)^{k-1}}{k}=-\frac{p ln p}{1-p}$

Step-by-step explanation:

The geometric distribution represents "the number of failures before you get a success in a series of Bernoulli trials. This discrete probability distribution is represented by the probability density function:"

$P(X=x)=(1-p)^{x-1} p$

Let X the random variable that measures the number os trials until the first success, we know that X follows this distribution:

$X\sim Geo (1-p)$

In order to find the expected value E(1/X) we need to find this sum:

$E(X)=\sum_{k=1}^{\infty} \frac{p(1-p)^{k-1}}{k}$

Lets consider the following series:

$\sum_{k=1}^{\infty} b^{k-1}$

And let's assume that this series is a power series with b a number between (0,1). If we apply integration of this series we have this:

$\int_{0}^b \sum_{k=1}^{\infty} r^{k-1}=\sum_{k=1}^{\infty} \int_{0}^b r^{k-1} dt=\sum_{k=1}^{\infty} \frac{b^k}{k}$ (a)

On the last step we assume that $0\leq r\leq b$ and $\sum_{k=1}^{\infty} r^{k-1}=\frac{1}{1-r}$ , then the integral on the left part of equation (a) would be 1. And we have:

$\int_{0}^b \frac{1}{1-r}dr=-ln(1-b)$

And for the next step we have:

$\sum_{k=1}^{\infty} \frac{b^{k-1}}{k}=\frac{1}{b}\sum_{k=1}^{\infty}\frac{b^k}{k}=-\frac{ln(1-b)}{b}$

And with this we have the requiered proof.

And since $b=1-p$ we have that:

$\sum_{k=1}^{\infty} \frac{p(1-p)^{k-1}}{k}=-\frac{p ln p}{1-p}$

4 0

3 years ago

A cube and the net for the cube are shown. <br><br> What is the surface area of this cube?

lozanna [386]

Ok, 6×25=150, since there are 6 sides, the answer is 150cm.

3 0

3 years ago

What is the standard form equation of the line shown below? Graph of a line going through negative 2, 3 and 1, negative 3

V125BC [204]

Answer:

3. 2x + y = −1

Step-by-step explanation:

To find the equation of the line, we write it first in the slope-intercept form:

$y=mx+q$

where

m is the slope

q is the y-intercept

From the graph, we see that the line crosses the y-axis at y = -1, so the y-intercept is -1:

$q=-1$

Now we have to find the slope, by calculating the rate of change of the line through 2 points:

$m=\frac{y_2-y_1}{x_2-x_1}$

Taking the two points at (-2,3) and (1,-3), we find:

$m=\frac{-3-3}{1-(-2)}=\frac{-6}{3}=-2$

So the equation of the line is

$y=-2x-1$

Now we have to re-arrange it in the standard form, so in the form

$ax+bx=c$

where a, b and c are integer numbers.

To do that, we simply add +2x on both sides of the equation of the line in the slope-intercept form, and we get:

$y+2x=-1$

So, option 3).

6 0

3 years ago

Mathematical connection A lighthouse casts a revolving beam of light as far as the pier

Alexandra [31]

The light's coverage area is given mathematically as i.e. Area= $\pi d^{2}$

Given that a lighthouse casts a revolving beam of light as far as they peir and asked the area is covered.

The revolving beam of light covers the whole space i.e entirely 360°

I.e. it forms a circle-like region

Area of circle = $\pi r^{2}$ { where r is the radius of circle}

Let's assume the distance traveled by light as "d"

as it forms a circular region, it forms a circle of radius "d"

Mathematical connection i.e. area of the region

⇒area of circular region= $\pi d^{2}$ {where d is the radius of region}

Learn more about area here:

brainly.com/question/27683633

#SPJ9

4 0

2 years ago

Please help<br> Me<br> Quick

Mrac [35]

Answer:

5

Step-by-step explanation:

First figure out the equation:

We know this is a linear equation because it appears to have a constant slope

Its y-intercept is where it passes in the y-axis, so it's -8

Furthermore, you should also figure out the slope. Notice that the y value increments by 3 every 1 x-value. Rise/Run is 3/1 = 3. Thus, the slope is 3

So your equation is g(x) = 3x - 8

Now we find our inverse function, which is just swapping the x-y values.

Thus, inverse of g(x) ==> y = 3x - 8 ==> x = 3y - 8 ==> x+8 = 3y ==> $\frac{x+8}{3}$

Plug in 7 for x and you get 5

5 0

3 years ago