Mean, Median and Mode

Median

Definition 111 (Median)

Let \(X\) be a continuous random variable with \(P D F f_X\). The median of \(X\) is a point \(c \in \mathbb{R}\) such that

\[ \int_{-\infty}^c f_X(x) d x=\int_c^{\infty} f_X(x) d x \]

Theorem 26 (Median from CDF)

The median of a random variable \(X\) is the point \(c\) such that

\[ F_X(c)=\frac{1}{2} \]

Mode

The mode is the peak of the PDF. We can see this from the definition below.

Definition 112 (Mode)

Let \(X\) be a continuous random variable. The mode is the point \(c\) such that \(f_X(x)\) attains the maximum:

\[ c=\underset{x \in \Omega}{\operatorname{argmax}} f_X(x)=\underset{x \in \Omega}{\operatorname{argmax}} \frac{d}{d x} F_X(x) \]

Note that the mode of a random variable is not unique, e.g., a mixture of two identical Gaussians with different means has two modes[Chan, 2021].

Mean

We have defined the mean as the expectation of \(X\). Here, we show how to compute the expectation from the CDF. To simplify the demonstration, let us first assume that \(X>0\).

Lemma 5 (Mean from CDF (X > 0))

Let \(X>0\). Then \(\mathbb{E}[X]\) can be computed from \(F_X\) as

\[ \mathbb{E}[X]=\int_0^{\infty}\left(1-F_X(t)\right) d t \]

Lemma 6 (Mean from CDF (X < 0))

Let \(X<0\). Then \(\mathbb{E}[X]\) can be computed from \(F_X\) as

\[ \mathbb{E}[X]=\int_{-\infty}^0 F_X(t) d t \]

Theorem 27 (Mean from CDF)

The mean of a random variable \(X\) can be computed from the CDF as

\[ \mathbb{E}[X]=\int_0^{\infty}\left(1-F_X(t)\right) d t-\int_{-\infty}^0 F_X(t) d t . \]

See Also

See chapter 4.4. of An Introduction to Probability for Data Science by Chan[Chan, 2021] for a more rigorous treatment of the mean, median and mode.

References and Further Readings

• Chan, Stanley H. “Chapter 4.4 Median, Mode, and Mean.” In Introduction to Probability for Data Science, 196-201. Ann Arbor, Michigan: Michigan Publishing Services, 2021.