Question

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?

Answer 1

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

Explanation:

`k=(-4)/2401`

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

`E(X)=56/15`

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

`E(X^2)=49/3`

Find the Variance:

`sigma^2=E(X^2)=[E(X)]^2=49/3-(56/15)^2=539/225`

Finally, standard deviation `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=(-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:)