How to calculate CDF of X, when X and P are given?

Question

Consider the probability space $(\mathbb{R},\mathcal{B}(\mathbb{R}),\mathbb{P})$, where $\mathbb{P}=0.1\delta_{-2}+0.7\delta_1+0.2\delta_{10}.$

If $X(\omega)=-2 I_{(-\infty,3]}(\omega)$, then the CDF of X is?

How to I calculate CDF?

Answer 1

My first answer was a mistake. Below a better answer:

The CDF of $X$ is the function $F(x) = \mathbb{P}(X\leq x)$.

We shall study first the function $f(x):=\mathbb{P}(X=x)$. The random variable $X$ only takes two values, either it is $-2$ or $0$.

Therefore $f(-2) = \mathbb{P} ((-\infty,3]) = 0.1+0.7=0.8$ (because $(-\infty,3]$ contains $-2$ and $1$), on the other hand $f(0) = \mathbb{P}((3,\infty))=0.2$.

Now $F(x)=P(X\leq x)$ since $X$ only takes the values $-2,0$ we conclude that $F(x)=0$ if $x<-2$, $F(x)=f(-2)=0.8$ when $-2\leq x <0$ and $F(x)=f(-2)+f(0)=0.8+0.2=1$ if $x\geq 0$.

Therefore $F(x)=0.8I_{[-2,\infty)}(x) + 0.2I_{[0,\infty)}(x)$