IB Stats

Definition of a probability density function: 

If f(x) is a probability density function, then $\displaystyle \int^6_0 f(x)\,dx = 1$.

This means for the probability density function in the question, $\displaystyle \int^6_0 Ax(6 - x)\,dx = 1$.

Now, notice that $\displaystyle \int^6_0 x(6- x)\,dx = 36$.

Therefore A = 1/36.

For part B, sketching a quadratic function should be easy.

For part C, just find $\displaystyle \int^6_4 f(x)\,dx$.

For part D, let W be the weight of a random jar. We can find $\mathbb E(W) = \displaystyle \int^6_0 xf(x)\,dx$ and $\mathbb E(W^2) = \displaystyle \int^6_0 x^2f(x)\,dx$ and​ then use the formula $\operatorname{Var}(W) = \mathbb E(W^2) - \left(E(W)\right)^2$

You should know how the standard deviation and the variance are related.