(Solved): (b) An elementary school assigns students (children) and parents (adults) to various tasks. Specifi ...

(b) An elementary school assigns students (children) and parents (adults) to various tasks. Specifically, a random task is assigned either 0 or 1 parents and either 0,1 or 2 children. (Note; it is possible that no one is assigned to a particular task; that is, one possibility is to have 0 parents and 0 children.) For this exercise, define: $\mathrm{X}=$ the number of adults assigned to a random task $\mathrm{Y}=$ the number of children assigned to a random task Overall, we have the following: - Most tasks do not require adult supervision. So, for a random task, $P(0$ adults assigned) $=.62$. - For a random task, $P(1$ child assigned $)=.338$ - For a random task, $\mathrm{P}(1$ or less children assigned $)=.705$ Page 2 of 6 - The conditional distribution on the number of children assigned to a task given that no adult is assigned to the task is: (i) Organize the above information into a probability tree model and assign the random variables as columns within the model. (ii) Determine $\rho (X,Y)$. Define $\mathrm{T}=$ the total number of people (i.e., adults and children together) assigned to a random task. (iii) Determine the distribution of $\mathrm{T}$ and compute Var(T) directly. (iv) Verify your answer to part (iii) by computing $\mathrm{Var}(X+Y)$.