Sums of independent random variablesAQA A-Level Further Maths: Mind map

Exam questions on Sums of independent random variables

1. $X$ and $Y$ are independent random variables with $E(X)=12$, $\text{Var}(X)=9$, $E(Y)=7$ and $\text{Var}(Y)=16$.
   A student claims that the standard deviation of $X+Y$ is $3+4=7$. Show that this is wrong and state the correct value.2 marks
2. The times, in minutes, taken to prepare and to cook a dish are modelled by independent random variables $X$ and $Y$, with $E(X)=8.5$, $\text{Var}(X)=1.44$, $E(Y)=12.3$ and $\text{Var}(Y)=2.56$. The total time is $T=X+Y$.
   Find the standard deviation of $T$, and state the assumption that makes it valid to add the variances.2 marks
3. The continuous random variable $X$ has probability density function $f(x)=\frac{x}{2}$ for $0\le x\le2$, and $f(x)=0$ otherwise. The discrete random variable $Y$ is the score when a fair six-sided die is rolled, so $E(Y)=\frac72$ and $\text{Var}(Y)=\frac{35}{12}$. $X$ and $Y$ are independent.
   Find $E(X+Y)$.3 marks