# What are the mean and standard deviation of the probability density function given by (p(x))/k=x^3-49x for  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?

Jan 27, 2016

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

#### Explanation:

$k = \frac{- 4}{2401}$

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

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

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

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

Find the Variance:

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

Finally, standard deviation $\sigma = \sqrt{{\sigma}^{2}} = \frac{7 \sqrt{11}}{15}$

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

hope that helped!

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