# Thread: Cumulative distribution function integration

1. ## Cumulative distribution function integration

The Question:
The continuous random variable X has probability density function given by:
$\displaystyle f(x)\ =\ \frac{1}{2}\ for\ x\ between\ 0\ and\ 1$
$\displaystyle \ \frac{3\ -\ x}{4}\ for\ x\ between\ 1\ and\ 3$

$\displaystyle \ 0\ otherwise$

a) Sketch the graph of f.
b) Explain why the value of n, the median of X, is 1.
c) Show that the value of u, the mean of X, is $\displaystyle \frac{13}{12}$.
d) Find P(X < 3u - n).

I've done a), b), and c), but am stuck on d).

My Attempt:
F(x) (the cumulative distribution function for X) is
$\displaystyle \int{\frac{3\ -\ x}{4}}\ from\ 1\ to\ x$, which becomes:
$\displaystyle [\frac{1}{4}(3x\ -\ x^2)]$ with limits 1 and x.

Substituting:
$\displaystyle \frac{3x\ -\ x^2}{4}\ -\ \frac{5}{8}$.

Now 3u - n = $\displaystyle \frac{9}{4}$.

Hence $\displaystyle F(\frac{9}{4})$ =:

$\displaystyle \frac{3(\frac{9}{4})\ -\ \(\frac{9}{4})^2}{4}\ - \frac{5}{8}$.

This works out to $\displaystyle \frac{55}{128}$.

However, the actual answer is $\displaystyle \frac{119}{128}$, and the cumulative distribution function F(x) is $\displaystyle 1\ -\ \frac{1}{8}(3\ -\ x)^2$.

Some advice on how to proceed would be appreciated.

isx99

2. Originally Posted by isx99
The Question:
The continuous random variable X has probability density function given by:
$\displaystyle f(x)\ =\ \frac{1}{2}\ for\ x\ between\ 0\ and\ 1$
$\displaystyle \ \frac{3\ -\ x}{4}\ for\ x\ between\ 1\ and\ 3$

$\displaystyle \ 0\ otherwise$

a) Sketch the graph of f.
b) Explain why the value of n, the median of X, is 1.
c) Show that the value of u, the mean of X, is $\displaystyle \frac{13}{12}$.
d) Find P(X < 3u - n).
$\displaystyle 3u-n=2.25$

$\displaystyle P(X< 3u-n)=P(X<2.25)=\int_0^{2.25} f(x) \; dx$

Now as $\displaystyle 1<2.25<3$ the integral may be written:

$\displaystyle \begin{array}{lll}P(X<2.25)&=&{\displaystyle \int_0^{1} f(x) \; dx+\int_1^{2.25} f(x) \; dx} \\ \\ &=&{\displaystyle \frac{1}{2}+\int_1^{2.25}\frac{3-x}{4}\;dx} \end{array}$

CB