(Solved): (a) X is a normally distributed random variable such that E(X)= and Var(X)=0.752. (i) Determine ...

(a) $\mathrm{X}$ is a normally distributed random variable such that $\mathrm{E}(\mathrm{X})=\alpha$ and $\mathrm{Var}(X)=0.75{\alpha }^2$. (i) Determine the probability that $\mathrm{X}$ takes on a value less than zero. (ii) Determine the probability that $\mathrm{X}$ takes on a value greater than half its expected value. (b) Suppose the time a construction crew needs to complete a building is distributed with mean 81 days and standard deviation 19 days. After completion, it takes additional time to install the plumbing and furnish the interior. Suppose the time needed for plumbing is distributed with mean 25 days and standard deviation 9 days, while the time needed for interior furnishing is distributed with mean 58 days and standard deviation 15 days. Further, the construction and plumbing times have a correlation coefficient of 0.83 , the construction and interior furnishing times have a correlation coefficient of 0.45 and the plumbing and interior furnishing times have a correlation coefficient of -0.26 . Let $\mathrm{T}$ be the total time from the beginning of building construction until the interior is furnished. Determine the expected value and standard deviation of $\mathrm{T}$.