The weight of boxes of type A is uniformly distributed on $[20,40]$.
As for boxes of type B, the weight $X$ gets values between $40$ and $60$ kilos and the density in that interval is $f_X=c\cdot x$, ($40<x<60$).
A company gets boxes of type A with possibility $60\%$ and of type B with possibility $40\%$.
- Determine the constant $c$.
- Under which weight $b_1$, a random box has possibility $20\%$ to weight less than $b_1$ ?
- Which is the mean value of the weight of a random box?
- If we load $100$ random boxes into a container, which is the possibility that the total weight of the loaded boxes is less than $39000$ kg ?
$$$$
I have done the following:
- We have that $$\int_{40}^{60}f_X\, dx=P(X\in [40,60])=40\% \Rightarrow c=0.0004$$ Is that correct?
- We are looking for $b_1$ such that $P(X<b_1)=20\%$, aren't we? Do we get that value as follows? \begin{align*}&P(X<b_1)=\int_{40}^{b_1}f_X\, dx \Rightarrow 0.2=\int_{40}^{b_1}0.0004x \Rightarrow 0.2=\left [0.0002x^2\right ]_{40}^{b_1} \\ & \Rightarrow 1000=\left [x^2\right ]_{40}^{b_1} \Rightarrow 1000=b_1^2-1600 \Rightarrow b_1^2=2600 \Rightarrow b_1=10\sqrt{26}\end{align*}
Could you give me also a hint for the last two questions?