# On Integrating the joint probability density function of two random variables

Suppose that the joint probability density function of two random variables $x$ and $y$ is given as $p(x,y)$. We know that the probability density function of $x$ can be found by integrating out $y$ i.e.

$$p(x) = \int_{-\infty}^{\infty} p(x,y)dy$$

While doing some calculation, I performed the following calculation by mistake.

$$Z = \int_{-\infty}^{y^*} p(x,y)dy$$

and I don't know what should I call $Z$. Is it the probability distribution function of $y$? If not, what is it that I calculated by mistake? My actual intention was to perform the integration on the entire real line.

• what is y* the limit of integration? Commented Feb 9, 2015 at 5:55
• $y^*$ is a deterministic variable and may also be treated as one of the realizations of $y$. Commented Feb 9, 2015 at 6:03

If the integral converges, then the result is a product of the probability density function for $$X$$ at $$x$$, and the conditional cumulative distribution function for $$Y$$ at $$y^\star$$ given $$X=x$$. This mixture will be a bivariate function so...
\qquad\begin{align}Z(x,y^\star) &= \int\limits_{-\infty}^{y^\star} p(x, \mathscr y)\,\mathrm d\mathscr y \\ &= p(x)\,\mathsf P(Y\leqslant y^\star\mid X=x)\end{align}
The expression $w:= p(x) Z$ is a real number, not a random variable. This value $w$ is known as the conditional probability that $Y \leq y^*$ given that $X=x$.