Need help answers please

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

Help please!!! ———————-

Marat540 [252]

B is the answer okay

5 0

3 years ago

Read 2 more answers

Andy joins a social networking site. After day three, he has 25 friends; after day eight, he has 40 friends. Write the equation

Allisa [31]

Answer:

Point slope intercept form: The equation for line is given by; $y-y_1=m(x-x_1)$ ......[1] ; where m is the slope and a point $(x_1, y_1)$ on the line.

Let x represents the number of days and y represents the number of friends.

As per the statement: After day three, he has 25 friends; after day eight, he has 40 friends.

⇒ We have two points i.e,

(3, 25) and (8, 40)

First calculate slope(m);

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

Substitute the given values we get;

$m = \frac{40-25}{8-3}=\frac{15}{5}$ = 3

now, substitute the given values of m=3 and a point (3, 25) in [1] we get;

$y-25=3(x-3)$

Using distributive property; $a \cdot(b+c) = a\cdot b + a\cdot c$

$y-25 = 3x - 9$

Add 25 on both sides, we get;

$y-25+25 = 3x - 9+25$

Simplify:

y =3x + 16

if x = 18 days, then;

y = 3(18) + 16 = 54+16 = 70

Therefore, he will have on day 18, if he continues to add the same number of friends each day is, 70 friends.

7 0

3 years ago

Triangle ABC has side a= 12, b=16, and c=c find the measure of angle a to the nearest whole number

Aleks04 [339]

C would equal 20. because 12 times its self plus 16 times its self equals 400. then I square 400 and got 20, which equals c

5 0

3 years ago

Find x, y, and z. Multiple choice answers are below!

labwork [276]

Answer:

The correct answer is B. x ≈ 4.5, y = 5, z = 9

Step-by-step explanation:

1. We will use the Pythagorean theorem for solving for x, this way:

x² = 6² - 4²

x² = 36 - 16

x² = 20

x = √20

x = 2 √5 = 4.47 ≈ 4.5

2. For solving for y, we will do it this way:

There are only two possible options: y = 5, included in answer B or y = 9 as it is defined on answer C.

If we review it carefully, we will realize that y = 9 is not accurate because as we can see in the graph y + 4 = z and in this specific case, 9 + 4 ≠ 5 but it's true that 5 + 4 = 9.

Because of this reason, the correct answer is B. x ≈ 4.5, y = 5, z = 9

7 0

3 years ago