Finding limits from cdf and expectation

Question

So I tried to find how each segment was formed in order to try and get an expression for expectation and solve from there, for example from c to 1, the probability is 0.2. From 1 to 2 it is also 0.2 and so on, except this seem to fall apart once I react the upper limit and thus can't get an expression, assuming this is even the correct approach, although I assume there is possibly a more simple one I'm missing?

Answer 1

What you seem to be missing is

• not that "from $c$ to $1$, the probability is $0.2$"
• but that when $c \le x \lt 1$ you have the probability that $X$ is less than or equal to $x$ is $0.2$ as this is a cumulative distribution function

Combine this for example with knowing that when $1 \le x \lt 2$ you have the probability that $X$ is less than or equal to $x$ is $0.4$, and it tell you that the probability that $X=1$ must be $0.4-0.2=0.2$

So you can calculate the probability that $X=c$, $1$, $2$, $3$ and so find $\mathbb E[X]$ in terms of $c$. Then solve to find the $c$ which makes $\mathbb E[X]=1.5$