Question

Suppose that the value of a stock varies each day from $12.82 to $28.17 with a uniform distribution.

Find the third quartile; 75% of all days the stock is below what value? (Enter your answer to the nearest cent.)

Answer 1

Let X be the value of the stock at any given day. X has a uniform distribution in the interval [12.82,28.17].

the pdf of X is

$\begin{align*} f(x)=\frac{1}{28.17-12.82}=\frac{1}{15.35}, \quad 12.82\le x\le 28.17 \end{align*}$

The CDF of X is

$\begin{align*} P(X<x)=F(x)&=\int_{12.82}^xf(x)dx\\ &=\int_{12.82}^x\frac{1}{15.35}dx\\ &=\left.\frac{x}{15.35}\right|_{12.82}^x\\ &=\frac{x-12.82}{15.35} \end{align*}$

that means

$\begin{align*} P(X<x)=\frac{x-12.82}{15.35} \end{align*}$

Let $Q_3$ be the value of the third quartile. The probability that the stock is below this value is 0.75

$\begin{align*} &P(X<Q_3)=0.75\\ \implies& \frac{Q_3-12.82}{15.35}=0.75\\ \implies& Q_3=12.82&plus;0.75\times 15.35\\ \implies& Q_3=24.33 \end{align*}$

The third quartile; 75% of all days the stock is below $24.33