Question: Suppose $X$ is a random variable with $\text{img}(X) = [0,10]$. If $\mathcal{P}(X > 5) \leq 0.4$ and $\mathcal{P}(X < 1) \leq 0.5$, what are the minimum and maximum possible values of $\mathbb{E}[X]$. I read that it's $0.5$ and $8.5$.
I attempted it but I didn't manage to find the best bounds.
Attempt: $\mathbb{E}[X] \leq \int_{0}^{1} x f(x) dx + \int_{5}^{10} x f(x)dx \leq \int_{0}^{1} x \mathcal{P}(X<1) dx + \int_{5}^{10} x \mathcal{P}(X > 5) dx = 15.25$ and in a similar way, I got $\mathbb{E}[X] \geq \int_{1}^{5} x (0.1) dx = 1.2$.
My bounds are not the best and the lower bound may as well be incorrect although I fail to see why. I am looking to find the best bounds and also verify if the bounds I calculated are correct (even if not the best).
Since $P(X<1)\leq 0.5,P(X>5)\leq 0.4,$ we must have $P(1\leq X\leq 5)\geq 0.1$.
In addition, $P(0\leq X\leq 10)=1$.
To minimize the mean, put as much mass at the lower end subject to these constraints by setting $$P(X=0)=0.5,P(X=1)=0.5\implies E[X]=0.5.$$
To maximize the mean, put as much mass at the higher end subject to these constraints by setting $$P(X=10)=0.4,P(X=5)=0.6\implies E[X]=7.$$