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Suppose $F$ is the CDF of a real valued random variable. I know that $F(- \infty) = 0$, because the RV cannot take a value less than that.

But I was thinking of an RV whose value for sure comes from, say, $[0, 1]$. Is it necessary that $F(0)$ be 0? More specifically, can I have some sort of a "point mass" at $x = 0$, so that the CDF is zero for $x < 0$, some number $k \in (0, 1)$ at 0 and then is continuous till it reaches 1 before or at 1?

What would be the associated PDF? I imagining a form like this: $\delta(x) + f(x)$, with $\delta$ being the Dirac-delta function. But how do I specify the exact value of $k$ on a PDF formulation? Is this even correct?

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  • $\begingroup$ Why does there have to be a PDF? $//$ How do you define a PDF? $\endgroup$
    – Dave
    Commented Jul 17 at 1:48
  • $\begingroup$ Doesnt have to be, just wanted to know if it's possible $\endgroup$
    – Ishan Kashyap Hazarika
    Commented Jul 17 at 7:14
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    $\begingroup$ Was a typo, should've been CDF. Edited it, thanks for pointing out $\endgroup$
    – Ishan Kashyap Hazarika
    Commented Jul 17 at 13:28
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    $\begingroup$ @AdamO I believe there's a universally understood definition for the CDF of a real-valued random variable $X$ and it's independent of the support or "event space;" namely, it's the function assigning to any real number $x$ the quantity $\Pr(X\le x).$ (There is a varying convention concerning whether the defining event is $X\le x,$ $X\lt x,$ or even $X\ge x$ or $X\gt x,$ but in every case the domain of the CDF is still $\mathbb R.$) $\endgroup$
    – whuber ♦
    Commented Jul 17 at 18:02
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    $\begingroup$ @whuber I think you are saying better what I was trying to express - I was moreover trying to chuckle at my own experience learning this. I think a great reference on this is McDonald and Weiss' "Real Anaylsis" text, Ch. 5. They establish elemental measure theory first, so that the probability axioms can be stated briefly and left at that. $\endgroup$
    – AdamO
    Commented Jul 17 at 18:26

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Consider the following probability function as an example:

$$f(x) = \left\{ \begin{array}{@{}ll@{}} 0.2, & \text{if}\ x=0, \\ 0.8 \cdot \beta(x;\alpha,\beta), & 0 < x < 1 \end{array}\right.$$

Here we have a probability mass of $0.2$ at $x=0$, i.e., the probability of seeing $x=0$ equals $0.2$. With (the remaining) probability $0.8$, $x \neq 0$; in this case, $x \sim \beta(\alpha,\beta)$, a Beta distribution with parameters $\alpha$ and $\beta$.

If we wanted to write this out in purely mathematical notation, we could use the indicator function $1(a)$, which equals $1$ if $a$ is true and $0$ otherwise:

$$f(x) = 0.2\cdot 1(x=0) + 0.8 \cdot \beta(x;\alpha,\beta)(1-1(x=0))$$

The cumulative distribution function would clearly be:

$$F(x) = \left\{\begin{array}{@{}ll@{}} 0, & x<0, \\ 0.2 + 0.8\cdot F_\beta(x;\alpha,\beta), &0 \leq x < 1 \\ 1, & 1 \leq x\\ \end{array}\right. $$

where $F_\beta(\cdot;\cdot)$ is the cumulative density function of the Beta distribution.

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