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 ?