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What is the formula for the expected number of successes in a binomial experiment with n trials and probability of success p? C

charle [14.2K]

Answer:

$(D)E[ X ] =np.$

Step-by-step explanation:

Given a binomial experiment with n trials and probability of success p,

$f(x)=\left(\begin{array}{c}n\\k\end{array}\right)p^x(1-p)^{n-x}, 0\leq x\leq n$

$E(X)=\sum_{x=0}^{n}xf(x)= \sum_{x=0}^{n}x\left(\begin{array}{c}n\\k\end{array}\right)p^x(1-p)^{n-x}$

Since each term of the summation is multiplied by x, the value of the term corresponding to x = 0 will be 0. Therefore the expected value becomes:

$E(X)=\sum_{x=1}^{n}x\left(\begin{array}{c}n\\x\end{array}\right)p^x(1-p)^{n-x}$

Now,

$x\left(\begin{array}{c}n\\x\end{array}\right)= \frac{xn!}{x!(n-x)!}=\frac{n!}{(x-)!(n-x)!}=\frac{n(n-1)!}{(x-1)!((n-1)-(x-1))!}=n\left(\begin{array}{c}n-1\\x-1\end{array}\right)$

Substituting,

$E(X)=\sum_{x=1}^{n}n\left(\begin{array}{c}n-1\\x-1\end{array}\right)p^x(1-p)^{n-x}$

Factoring out the n and one p from the above expression:

$E(X)=np\sum_{x=1}^{n}n\left(\begin{array}{c}n-1\\x-1\end{array}\right)p^{x-1}(1-p)^{(n-1)-(x-1)}$

Representing k=x-1 in the above gives us:

$E(X)=np\sum_{k=0}^{n}n\left(\begin{array}{c}n-1\\k\end{array}\right)p^{k}(1-p)^{(n-1)-k}$

This can then be written by the Binomial Formula as:

$E[ X ] = (np) (p +(1 - p))^{n -1 }= np.$

5 0

3 years ago

Justin walked his dog 5 miles in

Ad libitum [116K]

Answer:

Is the 5 separate or to gather like 45

3 0

3 years ago

Please help asap/ I need the correct answer for this homework ill give head?

nata0808 [166]

Answer:

Where is the question?

Step-by-step explanation:

please attach it...

8 0

3 years ago

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Describe the term linear pair, and give an example from the diagram.

Burka [1]

Answer/Step-by-step explanation:

A linear pair is formed when two straight lines intersect to form two angles that are adjacent to each other and are on a straight line. The sum of both adjacent angles equals 180°.

From the diagram given, examples of linear pair are:

<FEG and <GEN

<CDE and <ADC

These are some of the few examples we can see in the given diagram that are linear pairs.

6 0

2 years ago

An enterprise rents out paddle boats for all-day use on a lake. The owner knows that he can rent out all 40 of his paddleboats i

PSYCHO15rus [73]

Answer:

The owner's total revenue would be $204 if he charged $6 for each paddle boat rental.

Step-by-step explanation:

The owner has a total of 40 paddle boats and he can rent out all of them if he charges $3 per paddle boat. For each extra dollar, the number of paddle boats will decrease by 2. So, if he charges $6 (3 extra dollars) for each rental he can rent out:

40 - (3 x 2)

= 40 - 6

= 34 paddle boats.

The total revenue can be calculated by multiplying the number of paddle boats rented by the rent price per paddle boat.

Total Revenue = 34 x 6

= $204

The owner's total revenue would be $204 if he charged $6 for each paddle boat rental.

4 0

3 years ago

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