I am having problems while defining the PDF expression of a mixture distribution when some of its values are discrete. For example, imagine that a given random variable $\mathbb{X}$ takes values as follows:

\begin{equation} \mathbb{X} = \begin{cases} exp(1/\lambda),\quad \text{with probability}\,\, p\\ 0, \quad \text{with probability}\,\, (1-p) \end{cases} \end{equation}

So, my guess for the expression of the PDF of $\mathbb{X}$ is:

\begin{equation} f(x) = (1-p)\cdot \delta(x) + p\cdot \lambda e^{-\lambda\,x} \end{equation}

Is that correct? I am not sure about the $\delta(x)$.

Thank you very much in advance.

  • 1
    $\begingroup$ By definition, a discrete variable does not have a PDF. Therefore no mixture involving a discrete variable has a PDF, either. When you extend your concept of a "density" to include measures that are singular with respect to Lebesgue measure, such as the Dirac $\delta$, you can indeed express the distribution in this form--but most authorities seem to avoid calling this a "density." For an extended example of what $\delta$ is and how to work with it, see my answer at stats.stackexchange.com/a/73626/919. $\endgroup$ – whuber Jul 28 '17 at 13:41
  • $\begingroup$ Ok, thanks for the info, I will read it carefully. Anyway, shall I understand that I can express the distribution in that way? $\endgroup$ – Gabriel Jul 28 '17 at 13:58
  • $\begingroup$ Pretty close. To be clear and rigorous, it is essential that you multiply the second term by the indicator function of its intended support. $\endgroup$ – whuber Jul 28 '17 at 14:01
  • $\begingroup$ Something like this? $f(x) = (1-p)\cdot \delta(x) + p\cdot \lambda e^{-\lambda\,x}\cdot H(x)$ where H(x) is the Heaviside step function? $\endgroup$ – Gabriel Jul 28 '17 at 16:00
  • $\begingroup$ Yes, that would work. $\endgroup$ – whuber Jul 28 '17 at 16:36

To sum up:

Yes, you can express it as:

\begin{equation} f(x) = (1-p)\cdot \delta(x) + p\cdot \lambda e^{-\lambda\,x} \cdot H(x) \end{equation}

Note that Heaviside step function has been included in the second term so as to be rigorous and specify the indicator function of the intended support for this term, as suggester by @whuber

See the comments for more info.


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