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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.

\begin{equation} p(x) = \int_{-\infty}^{\infty} p(x,y)dy \end{equation}

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

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

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.

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  • $\begingroup$ what is y* the limit of integration? $\endgroup$
    – Henry
    Commented Feb 9, 2015 at 5:55
  • $\begingroup$ $y^*$ is a deterministic variable and may also be treated as one of the realizations of $y$. $\endgroup$
    – user146290
    Commented Feb 9, 2015 at 6:03

2 Answers 2

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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...

$\qquad\begin{align}Z(x,y^\star) &= \int\limits_{-\infty}^{y^\star} p(x, \mathscr y)\,\mathrm d\mathscr y \\ &= p(x)\,\mathsf P(Y\leqslant y^\star\mid X=x)\end{align}$

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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 $X=x$.

A reference is equation (8.16) in Klenke, Probability theory, e.g. in google books. If an example is preferred, then Conditional probabilities from a joint density function is a good one.

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