# Find the distribution parameters

On STAT101 your assessment is based on: Final Exam 50% Learn based on-line assessment 30% Assignments 20% Consider three random variables $X$, $Y$ and $Z$ which respectively represent the exam, on-line assessment total and assignment scores (out of 100%) of a randomly chosen student. Assume that $X$, $Y$ and $Z$ are independent (this is clearly not true, but the answers may be a reasonable approximation). Suppose that past experience suggests the following properties of these assessment items (each out of 100%): $E \left ( X \right ) = 61$, $sd(X) = 20$, $E(Y) = 72$, $sd(Y) = 22$ and $E(Z) =65$, $sd(Z) = 24$.

a) Find the distribution parameters, $E(T)$ and $Var(T)$, for the total mark, $T$, where: $T = 0.5X+0.3Y+0.2Z$.

## 1 Answer

Here are some hints that will hopefuly guide you to the answers.

The expectation $E(T)$ can be obtained using the linearity properties of the expected value.

The variance $\mbox{var}(T)$ can be obtained using the expression of the variance of a sum of random variable and using the independence of $X$, $Y$, and $Z$ to determine the value of the covariance terms that appear.