Question

Say we have 14 people in this room; it was found there are 4 people having a height from
150 to 160 cm; 4 people having a height from 160 to 170 cm; 6 people having a height from 170 to 180 cm; what is the estimated height average in the class?

Answer 1

Let $X$ be a random variable such that $X(p)=\text{height of $p$}$ for every pupil $p$ in the room.

We have $P(X\in[150,160])=\frac{4}{14}=\frac{2}{7}$, $P(X\in[160,170])=\frac{4}{14}=\frac{2}{7}$, $P(X\in[170,180])=\frac{6}{14}=\frac{3}{7}$

Alos suppose that there is an uniform distribution on $[150,160]$, $[160,170]$ and $[170,180]$.

So $\rho_X(x)=\begin{cases} 0,&\text{if $x\not\in[150,180]$}\\ \frac{1}{35},&\text{if $x\in[150,160)$}\\ \frac{1}{35},&\text{if $x\in[160,170)$}\\ \frac{3}{70},&\text{if $x\in[170,180]$} \end{cases}=$

$=\begin{cases} 0,&\text{if $x\not\in[150,180]$}\\ \frac{1}{35},&\text{if $x\in[150,170)$}\\ \frac{3}{70},&\text{if $x\in[170,180]$} \end{cases}$, where $\rho_X$ is probability density function of $X$.

So average height is $EX=\int\limits_{\mathbb R}x \rho_X(x)dx=\int\limits_{150}^{170}\frac{1}{35}xdx+\int\limits_{170}^{180}\frac{3}{70}xdx=$

$=\frac{1}{70}x^2\bigl|_{150}^{170}+\frac{3}{140}x^2\bigl|_{170}^{180}=\frac{1165}{7}$

Answer: $\frac{1165}{7}$ cm