I am given two random variables $X$ and $Y$, both are independent:
and X distributioN:
-2 -1 3
0.2 0.3 0.5
and Y distribution:
-1 0 2
0.2 0.4 0.4
I have to create distribution and then I have to calculate expected value and variance of $3X – 2Y$, $X^2 + Y^2$, $X \cdot Y$.
I do not understand how to create distribution of $3X – 2Y$, $X^2 + Y^2$ and $X \cdot Y$ accordingly. After creating distribution, I can calculate Expected value and variance, so I think I do not need any special formulas. However, I cannot understand how $3X – 2Y$ and others should be calculated.
Should I take "each $X$" and "each $Y$" and calculate $3X – 2Y$, so there would be 36 items?
I do not really understand how this works and what is a meaning of $3X – 2Y$ if $X$ and $Y$ are two random indpendent variables.
Best Answer
I'm assuming that you meant $X$ and $Y$ to be independent such that $X$ has probability mass function given by $$ P(X=k)= \begin{cases} 0.5 &\quad \text{if }k=3\\ 0.3 &\quad \text{if }k=-1\\ 0.2 &\quad \text{if }k=-2\\ \end{cases} $$ and $Y$ has probability mass function $$ P(Y=k)= \begin{cases} 0.4 &\quad \text{if }k=2\\ 0.4 &\quad \text{if }k=0\\ 0.2 &\quad \text{if }k=-1\\ \end{cases} $$ Then using the law of the unconscious statistician we obtain $$ E[X]=3\cdot 0.5+(-1)\cdot 0.3+(-2)\cdot 0.2=0.8 $$ and similarly $E[Y]=0.6$. By applying the law again, we can also obtain expressions for $E[X^2]$ and $E[Y^2]$. Having them in hand, we can find the variance of $X$ and $Y$ by the formula $$ \mathrm{Var}(X)=E[X^2]-E[X]^2 $$ and similarly for $Y$.
Now you're able to calculate the mean and variance of $3X-2Y$ by using that $$ E[3X-2Y]=3E[X]-2E[Y], $$ and $$ \mathrm{Var}(3X-2Y)=3^2\mathrm{Var}(X)+(-2)^2\mathrm{Var}(Y) $$ where we explicitly used that $X$ and $Y$ are independent, see e.g. this.
In a similar manner you can compute the mean and variance of $X^2+Y^2$, but to compute the mean and variance of $X\cdot Y$ you need to use that $$ E[g(X)g(Y)]=E[g(X)]\cdot E[g(Y)] $$ for any (measurable) function $g$.