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3 years ago

7

emma walked 1/4bmile on Monday, 2/4 mile on tuesday, and 3/4 mile on Wednesday. If the pattern continues, how many miles will sh

e walk on Friday? Explain how you found the numbers of miles

1 answer:

frosja888 [35]3 years ago

5 0

Monday 1/4

Tuesday 2/4

Wednesday 3/4

Thursday 4/4

Friday 5/4 or 1 1/4 miles

You just add 1/4 each day.

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

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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.

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

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