# Category Archives: probability density function

## Geometric probability

in each case, consider a point picked uniformly randomly from the interior of  the region. Find the probability density function for the $$x$$-coordinate.

1. The square with corner : $$(-2,0), (0,2), (2,0), (0,-2)$$
2. The triangle with corners: $$(-2,0), (1,0), (0,2)$$
3. The polygon with corners: $$(0,2),(2,1), (1,-1), (-1,0)$$

[Pitman p277, # 12]

## conditional densities

Let $$X$$ and $$Y$$ have the following joint density:

$f(x,y)=\begin{cases}2x+2y -4xy & \text{for } 0 \leq x\leq 1 \ \text{and}\ 0 \leq y \leq 1\\ 0& \text{otherwise}\end{cases}$

1. Find the marginal densities of $$X$$ and $$Y$$
2. find $$f_{Y|X}( y \,|\, X=\frac14)$$
3. find $$\mathbf{E}(Y \,|\, X=\frac14)$$

[Pitman p426 # 2]

## geometric probability: marginal densities

Find the density of the random variable $$X$$ when the pair $$(X,Y)$$ is chosen uniformly from the specified region in the plane in each case below.

1. The diamond with vertices at $$(0,2), (-2,0), (0,-2), (2,0)$$.
2. The triangle with vertices $$(-2,0), (1,0), (0,2)$$.

[Pitman p 277, #12]

## probability density example

Suppose  $$X$$ takes values in$$(0,1)$$ and has a density

$f(x)=\begin{cases}c x^2 (1-x)^2 \qquad &x\in(0,1)\\ 0 & x \not \in (0,1)\end{cases}$

for some $$c>0$$.

1. Find $$c$$.
2. Find $$\mathbf{E}(X)$$.
3. Find $$\mathrm{Var}(X)$$.

## Infinite Mean

Suppose that $$X$$ is a random variable whose density is

$f(x)=\frac{1}{2(1+|x|)^2} \quad x \in (-\infty,\infty)$

1. Draw a graph of $$f(x)$$.
2. Find $$\mathbf{P}(-1 <X<2)$$.
3. Find $$\mathbf{P}(X>1)$$.
4. Is $$\mathbf{E}(X)$$ defined ? Explain.

## Car tires

The air pressure in the left and right front tires of a car are random variables $$X$$ and $$Y$$, respectively. Tires should be filled to 26psi. The joint pdf is

$$f(x,y) = K(x^2+y^2), \quad 20 \leq x,y \leq 30$$

1. What is $$K$$ ?
2. Are the random variables independent ?
3. What is the probability that both tires are underfilled ?
4. What is the probability that $$|X-Y| \leq 3$$ ?
5. What are the marginal densities ?