# cdf of a “mixed” random variable

If $$B$$ is a random variable following a known distribution (in my case $$B$$ follows a Pareto distribution with parameters $$\alpha = 1.5$$ and $$\lambda = 1000$$) and $$R$$ is independent of $$B$$ following a normal distribution of mean $$0.08$$ and standard deviation $$0.2$$. Is there any way to find the cdf of $$X=B \times 1_{\lbrace R \leq -0.1 \rbrace}$$. I tried a lot (by conditionning on the variable $$R$$) and it did not lead me to somewhere. Any hints?

• The distribution of $1_{\{R\,\le\,-1\}}$ is Bernoulli with expected value about $0.1586553$. No information about the distribution of $R$ beyond that is relevant here. ${}\qquad{}$ – Michael Hardy Oct 2 '15 at 0:29

Let $\{A = 0\} = \{R\le -0.1\}$, and $\{A = 1\} = \{R > -0.1 \}$.
Let $P(A = 1) = p$
where $u(x)$ is the step function. You can play with the value of $u(0)$ to make the CDF left or right continuous, whatever your convention may be.