Mathematics, 15.10.2019 15:10 ousmaner

Identify the expression for calculating the mean of a binomial distribution.

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Answer from: TPug24

A random variable

following a binomial distribution with success probability

across

trials has PMF

$\mathbb P(X=x)=\begin{cases}\dbinom nxp^x(1-p)^{n-x}&\text{for }x\in\{0,1,\ldots,n\}\\\\0&\text{otherwise}\end{cases}$

where $\dbinom nx=\dfrac{n!}{x!(n-x)!}$ .

The mean of the distribution is given by the expected value which is defined by

$\mathbb E(X):=\displaystyle\sum_xx\mathbb P(X=x)$

where the summation is carried out over the support of

. So the mean is

$\displaystyle\sum_{x=0}^nx\binom nxp^x(1-p)^{n-x}$

Because this is a proper distribution, you have

$\displaystyle\sum_x\mathbb P(X=x)=1$

which is a fact that will be used to evaluate the sum above.

$\displaystyle\sum_{x=0}^nx\binom nxp^x(1-p)^{n-x}$
$\displaystyle\sum_{x=1}^nx\binom nxp^x(1-p)^{n-x}$
$\displaystyle\sum_{x=1}^n\frac{xn!}{x!(n-x)!}p^x(1-p)^{n-x}$
$\displaystyle np\sum_{x=1}^n\frac{(n-1)!}{(x-1)!(n-x)!}p^{x-1}(1-p)^{n-x}$
$\displaystyle np\sum_{x=1}^n\frac{(n-1)!}{(x-1)!((n-1)-(x-1))!}p^{x-1}(1-p)^{(n-1)-(x-1)}$

Letting y=x-1

, this becomes

$\displaystyle np\sum_{y=0}^{n-1}\frac{(n-1)!}{y!((n-1)-y)!}p^y(1-p)^{(n-1)-y}$

Observe that the remaining sum corresponds to the PMF of a new random variable

which also follows a binomial distribution with success probability

, but this time across n-1

trials. Therefore the sum evaluates to 1, and you're left with

as the expression for the mean for

Answer from: Quest

2 & 7

step-by-step explanation:

he climbs 300 ft/hr at a constant rate. the peak is at 1,500 ft.

when going up, he reaches 600 ft in 2 hours. remember, that he is going in a constant rate. divide 600 with 300:

600/300 = 2

assuming that he takes the same route back (meaning that it's a 1500 ft route back, you can solve.

note also that you are solving when he gets back to 600 ft after he reaches the peak.

first, find the amount of hours it takes to get up the peak. divide the total amount (1500), with the ft/her (300) to get the amount of hours (x):

1500/300 = x

x = 5

it takes jamie 5 hours to get up. now, he is returning down, and reaches 600 (his current destination). remember that he climbs at a constant 300 ft/hr. divide 600 with 300:

600/300 = x

x = 2

now add the two numbers together.

x = 5 + 2 = 7

jamie takes 7 hours to go up the mountain and reach his current position when descending.

(total amount/constant rate) = (time).

remember that it is a constant rate, so in total, when he gets back down the mountain, it will take the same amount of time as when he goes up. the total amount for the round trip is: 5 + 5 = 10.

Answer from: Quest

imma say 16%. sorry if its wrong. i'm doing the same stuff right now

step-by-step explanation:

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