What are the mean and standard deviation of the probability density function given by $\frac{p (x)}{k} = x^{3} - 49 x$ $(p(x))/k=x^3-49x$ for $x \in [0, 7]$ $x in [0,7]$ , in terms of k, with k being a constant such that the cumulative density across the range of x is equal to 1?

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What are the mean and standard deviation of the probability density function given by $\frac{p (x)}{k} = x^{3} - 49 x$ $(p(x))/k=x^3-49x$ for $x \in [0, 7]$ $x in [0,7]$ , in terms of k, with k being a constant such that the cumulative density across the range of x is equal to 1?

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Answer 1 · 2016-01-27T00:52:50

Integrate p(x) over $[0, 7]$ $[0,7]$ , set equal to 1, then solve for k ...

Explanation:

$k = \frac{- 4}{2401}$ $k=(-4)/2401$

Next, using k above, integrate $x f (x)$ $xf(x)$ over $[0, 7]$ $[0,7]$ to find $E (X)$ $E(X)$

$E (X) = \frac{56}{15}$ $E(X)=56/15$

Now, using k above, integrate $x^{2} f (x)$ $x^2f(x)$ over $[0, 7]$ $[0,7]$ to find $E (X^{2})$ $E(X^2)$

$E (X^{2}) = \frac{49}{3}$ $E(X^2)=49/3$

Find the Variance:

$σ^{2} = E (X^{2}) = {[E (X)]}^{2} = \frac{49}{3} - {(\frac{56}{15})}^{2} = \frac{539}{225}$ $sigma^2=E(X^2)=[E(X)]^2=49/3-(56/15)^2=539/225$

Finally, standard deviation $σ = \sqrt{σ^{2}} = \frac{7 \sqrt{11}}{15}$ $sigma=sqrt(sigma^2)=(7sqrt11)/15$

If you really need to know the mean and standard deviation in terms of k , then simply divide these parameters by $k = \frac{- 4}{2401}$ $k=(-4)/2401$ , then multiply each by the variable $k$ $k$ .

hope that helped!

Note: Used the Solver feature of my TI-84 to find all these values:)

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What are the mean and standard deviation of the probability density function given by $\frac{p (x)}{k} = x^{3} - 49 x$ $(p(x))/k=x^3-49x$ for $x \in [0, 7]$ $x in [0,7]$ , in terms of k, with k being a constant such that the cumulative density across the range of x is equal to 1?

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Explanation:

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What are the mean and standard deviation of the probability density function given by p(x)k=x3−49x(p(x))/k=x^3-49x for x∈[0,7] x in [0,7], in terms of k, with k being a constant such that the cumulative density across the range of x is equal to 1?

1 Answer

Explanation:

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What are the mean and standard deviation of the probability density function given by $\frac{p (x)}{k} = x^{3} - 49 x$ $(p(x))/k=x^3-49x$ for $x \in [0, 7]$ $x in [0,7]$ , in terms of k, with k being a constant such that the cumulative density across the range of x is equal to 1?