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Suppose

$f_X(x) = \left\{\begin{matrix} 0.5 & x = 0 \\ x & 0 < x \le 1 \end{matrix}\right.$

What is the MODE of this distribution? I think it should be $0$ coz it has the highest mass, even though $1$ has the highest density.

Another one

$f_X(x) = \left\{\begin{matrix} x & 0 < x \le 1 \\ 1 & 1 < x \le 1.5 \end{matrix}\right.$

What would be the MODE in this case? $1?~~ 1.5?$ or any number in $[1, 1.5]$

Many thanks in advance

ethanol
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2 Answers2

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An element x∈S that maximizes the probability density function f is called a mode of the distribution.

In the first case, it is x = 1 and in the second case, it is any number between [1,1.5].

Satish Ramanathan
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  • But first one is a mixed distribution, which has both PMF and PDF. So even though $1$ has the highest density, most popular value is $0$ with prob. $0.5$. So shouldn't the mode be $0$? – ethanol Feb 14 '15 at 18:00
  • Why do you think it is a mixed distribution, a pdf can be expressed as a value at one point and another function at a different interval. I would consider the first case as a pdf defined in two intervals. – Satish Ramanathan Feb 14 '15 at 18:18
  • If we consider it as a pdf in different intervals, you have to prove that $\int_{-\infty}^{1} f(x) dx = 1$ , Now $\int_{0}^{1} f(x)dx = 0.5 + \int_{0}^{1}xdx = 0.5 + x^2/2|_0^{1} = 0.5 + 0.5 = 1$ and hence you can consider it to be a probability density function – Satish Ramanathan Feb 14 '15 at 18:29
  • @SatishRamanathan in second case also, maximum value of PDF can be 1 only, how it can greater than 1? – Ankita Goyal Jan 26 '23 at 18:47
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This is a bit controversial. If you go according to the definition of mode. It is the value of the random variable with the highest probability.

If you compare a point (with non-zero probability) from a discrete distribution to that of a continuous distribution, you will clearly notice that probability at a point for the discrete distribution would be non-zero.

Though, the probability of the continuous distribution at a point would be zero. (Because the probability is spread over infinite values inside the range, with some points having more density than the others.)

Following that logic, $x=0$ would be the mode of your first question.

For the second part of your question, this is a matter of the mode not being clearly defined.

Ask yourself the question. What is the mode from the following discrete distribution? $\{1,2,3,3,4,5,5\}$

Is it $3$ or $5$? Similar is the ambiguity in the second question at hand.

Hursh
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