"Define a probability measure $\mu$ on $(\Bbb{R},\mathcal{B}(\Bbb{R}))$ by:
$\mu = 0.3\delta_0 + 0.4\delta_1 + 0.3\delta_4$
Find the distribution function $F$ of $\mu$"
This is a question from a previous exam paper, and I am trying to get to grips with questions like these. In my notes I have the Cumulative distribution function defined as:
$F_X(t) = \Bbb{P}({X\leq t})$
I also have the dirac measure as:
$\delta_x(A) \ = $$\left\lbrace \matrix{1,\ x\in A\cr 0, \ x \notin A} \right\rbrace$
I've also read something about the dirac measure being 0 at all points apart from $x=0$ where it equals 1, but it's just confusing me even more
Could anyone help?
Thank you
Your formula for the CDF is correct, it remains to write it as a function of the PDF: $$ F_X(t)=\mu((-\infty,t])=\begin{cases} 0 &\text{ if }0\notin(-\infty,t], 1\notin(-\infty,t],\text{ and }4\notin(-\infty,t],\\ 0.3 &\text{ if }0\in(-\infty,t], 1\notin(-\infty,t],\text{ and }4\notin(-\infty,t],\\ 0.3+0.4&\text{ if }0\in(-\infty,t], 1\in(-\infty,t],\text{ and }4\notin(-\infty,t],\\ 0.3+0.4+0.3&\text{ if }0\in(-\infty,t], 1\in(-\infty,t],\text{ and }4\in(-\infty,t].\\ \end{cases} $$ Therefore, $$ F_X(t)=\begin{cases} 0&\text{ if }t<0,\\ 0.3 &\text{ if }0\le t<1,\\ 0.7&\text{ if }1\le t<4,\\ 1&\text{ if }t\ge4.\\ \end{cases} $$