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Anika [276]
3 years ago
12

Find the mean, variance &a standard deviation of the binomial distribution with the given values of n and p.

Mathematics
1 answer:
MrMuchimi3 years ago
7 0
A random variable following a binomial distribution over n trials with success probability p has PMF

f_X(x)=\dbinom nxp^x(1-p)^{n-x}

Because it's a proper probability distribution, you know that the sum of all the probabilities over the distribution's support must be 1, i.e.

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

The mean is given by the expected value of the distribution,

\mathbb E(X)=\displaystyle\sum_xf_X(x)=\sum_{x=0}^nx\binom nxp^x(1-p)^{n-x}
\mathbb E(X)=\displaystyle\sum_{x=1}^nx\frac{n!}{x!(n-x)!}p^x(1-p)^{n-x}
\mathbb E(X)=\displaystyle\sum_{x=1}^n\frac{n!}{(x-1)!(n-x)!}p^x(1-p)^{n-x}
\mathbb E(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)}
\mathbb E(X)=\displaystyle np\sum_{x=0}^n\frac{(n-1)!}{x!((n-1)-x)!}p^x(1-p)^{(n-1)-x}
\mathbb E(X)=\displaystyle np\sum_{x=0}^n\binom{n-1}xp^x(1-p)^{(n-1)-x}
\mathbb E(X)=\displaystyle np\sum_{x=0}^{n-1}\binom{n-1}xp^x(1-p)^{(n-1)-x}

The remaining sum has a summand which is the PMF of yet another binomial distribution with n-1 trials and the same success probability, so the sum is 1 and you're left with

\mathbb E(x)=np=126\times0.27=34.02

You can similarly derive the variance by computing \mathbb V(X)=\mathbb E(X^2)-\mathbb E(X)^2, but I'll leave that as an exercise for you. You would find that \mathbb V(X)=np(1-p), so the variance here would be

\mathbb V(X)=125\times0.27\times0.73=24.8346

The standard deviation is just the square root of the variance, which is

\sqrt{\mathbb V(X)}=\sqrt{24.3846}\approx4.9834
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Answer:

Step-by-step explanation:

Slope of line A = \frac{\text{Rise}}{\text{Run}}

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Slope of line B = \frac{9}{6}

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Slope of line C = \frac{6}{8}

                         = \frac{3}{4}

5). Slope of the hypotenuse of the right triangle = \frac{\text{Rise}}{\text{Run}}

                                                                                = \frac{90}{120}

                                                                                = \frac{3}{4}

Since slopes of line C and the hypotenuse are same, right triangle may lie on line C.

6). Slope of the hypotenuse = \frac{30}{10}

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Therefore, this triangle may lie on the line A.

7). Slope of hypotenuse = \frac{18}{24}

                                        = \frac{3}{4}

Given triangle may lie on the line C.

8). Slope of hypotenuse = \frac{21}{14}

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Given triangle may lie on the line B.

9). Slope of hypotenuse = \frac{36}{24}

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Given triangle may lie on the line B.

10). Slope of hypotenuse = \frac{48}{16}

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2 years ago
What is 15% of 60 show your work to prove your answer
Mila [183]
<span>To convert a percentage to a decimal you move the decimal two places to the left. Therefore:

15.% becomes .15

You then multiply this decimal to the number you are trying to find the percentage of.

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</span>
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2 years ago
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dmitriy555 [2]

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<u>Answer</u><u>:</u>

<u>{303}^{18}</u>

<u>Hope </u><u>you</u><u> could</u><u> get</u><u> an</u><u> idea</u><u> from</u><u> here</u><u>.</u>

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GenaCL600 [577]

Answer:

24ft

Step-by-step explanation:

Pythagorean Theorem!!

a^2 + b^2 = c^2

------------------------

In this case, the ladder is c, and the base is b.

------------------------

Plug the numbers in the formula.

a^2 + 7^2 = 25^2

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Solve.

a^2 + 49 = 625

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\sqrt{a^2} = \sqrt{576}

a = 24

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