Was wondering if anyone could help me out. So far I've used the fact that the density function is always equal to 1. So I have done the integral of the density function between 0 and 1 and worked out my first equation in the steps of finding the value for the constants. Then I used the fact what we are given the value of the Expected Value which is the same as the integral between 0 and 1 of the density function multiplied by 'x'. Which has given me a second equation to figure out the value of the constants. Then I compared the equation to get the answer that a=(3/2) and b=(1/2). Now that I have these values, I know that I have to do an integral of the new density function, but I don't know what boundaries to set. I was assuming it would be 1/2 and + infinity since the question just asks for x to be greater than a half, but I don't know if this is correct.
You want $$\mathbb P(0.2 \lt X)=\int\limits_{0.2}^{+\infty} f_X^{\,}(x)\, dx =\int\limits_{0.2}^1 \left(\frac32 x^2 +\frac12\right)\, dx$$
as the lower limit (from the question) is $0.2$ and upper limit (from the support) is $1$