# What is the correct way of expressing this probability mass function?

Suppose I have a discrete random variable $$S$$. Now further suppose that I have a random variable $$X$$ which is a continuous random variable on the real line. I wanted to express the p.m.f. $$P[S=s]$$ for $$S$$ using the expression: $$$$P[S=s]= \int_{\mathbb{R}}P[S=s\mid x]P[x\mid s]dx$$$$

A professor I spoke to mentioned that if both $$S$$ and $$X$$ were discrete, this expression would have been correct, however, given that $$X$$ is a continuous random variable, the way I have expressed it implies a zero probability (I suppose because of $$P[x\mid s]$$). I was told to express this correctly, I have to express it in terms of probability measures.

I have been spending the better part of yesterday on Shiryaev's probability book and on probability measures and Lebesgue integrals; but while I know he is correct about my incorrect expression, I am not sure exactly how this needs to be expressed in terms of probability measures.

• Not sure what you're trying do there. But I guess you need conditional density. Commented Jul 3, 2023 at 8:58

It sounds like you are trying to write the law of total probability. For a discrete random variable $$S$$ and a continuous conditioning random variable $$X$$ with marginal density $$f_X$$, you can write this as:

$$\mathbb{P}(S=s) = \int \limits_\mathbb{R} \mathbb{P}(S=s|X=x) f_X(x) \ dx.$$

Observe that the integrand here consists of the conditional probability mass for $$S|X$$ multiplied by the marginal density of $$X$$ (not the conditional density). I see no reason you would need to write this in terms of the probability measures or the Lebesgue integral (since you know that $$X$$ is continuous here), but if you really want to do so then you can take the probability measure $$\mu_X$$ or the distribution function $$F_X$$ for $$X$$ and write the relevant result as:

$$\mathbb{P}(S=s) = \int \limits_\mathbb{R} \mathbb{P}(S=s|X=x) \ d F_X(x) = \int \limits_\mathbb{R} \mathbb{P}(S=s|X=x) \ d \mu_X.$$

• Hi Ben. You are correct. However, I noticed I made an error in formulating the question, which explains the unnecessary conditioning in the second term. My intended question was, suppose there $S$ AND $I$ are the discrete random variables, and I wish to express $P[S=s\mid I=i]=\int_{\mathbb{R}}P[S=s\mid I=i,X=x]P[X=x \mid I=i]dx$, where as before $X$ is continuous. How would the formulation would this specific case be?
– Carl
Commented Jul 3, 2023 at 23:45
• Probably best to ask this as a new question.
– Ben
Commented Jul 4, 2023 at 0:07