The random variable \(X\) is normally distributed with mean \(\mu\), as shown in the diagram. The shaded region between \(7\) and \(\mu\) represents \(45\%\) of the distribution.
Find \(\mathrm{P}(X<7)\).[1]
The standard deviation of \(X\) is \(1.8\).
Find the value of \(\mu\).[2]
The random variable \(Y\) has the distribution \(\mathrm{N}(\lambda,2.8^2)\). The random variables \(X\) and \(Y\) are independent, and \(\mathrm{P}((X>7)\cap(Y>7))=0.38\).
Find the value of \(\lambda\).[3]
Given that \(\mathrm{P}(Y>a)=2\mathrm{P}(Y>7)\), find the value of \(a\).[2]