Mode of a Mixed Distribution

Asked Jan 26 '23 at 19:47

Active Jan 26 '23 at 19:47

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$X$ has the following mixed distribution:

$P[0] = 0.2$ $$f(x) = \left\{ \begin{array}{cl} \frac{0.5xe^{\frac{-x}{100}}}{100} & x > 0 \\ 0 & \text{Otherwise} \\ \end{array} \right.$$

Calculate the mode of $X$ .

I think the answer should be $0$ as the maximum value for the $f(x)$ occurs at the point $x=100$ . Here, $f(100) = 0.18 (approx.) < 0.2$ , i.e less than the value which occurs at $0$.

However, the answer is $100$ only.

MODE of mixed distribution The same kind of question has been answered here but there is no clear cut solution witch which I can agree. So, any help would be appreciated.

asked Jan 26 '23 at 19:47

Ankita Goyal

The problem is that many of the references define "mode" in terms of a discrete distribution or a continuous distribution. This is a mixture, so what to do? – mjw Jan 26 '23 at 21:13
What is the source of your question? – callculus42 Jan 27 '23 at 01:47
This question came in one exam of mine. – Ankita Goyal Jan 27 '23 at 05:14
@mjw mode is the value that maximizes the distribution so here max value occurs at 0, that's what I thought but I am not sure. – Ankita Goyal Jan 27 '23 at 05:16
Well, that would be a (weighted) delta function at $x=0$, right? $\lim_{\epsilon\to 0} \int_{0^-}^\epsilon f(x) , dx = 0.2.$ – mjw Jan 27 '23 at 07:21
I didn't understand this. – Ankita Goyal Jan 27 '23 at 07:25
The delta function is explained here: https://mathworld.wolfram.com/DeltaFunction.html – mjw Jan 30 '23 at 00:03

Mode of a Mixed Distribution

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