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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.
- The square with corner : \( (-2,0), (0,2), (2,0), (0,-2) \)
- The triangle with corners: \( (-2,0), (1,0), (0,2) \)
- 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:
\[ f(x,y)=\begin{cases}2x+2y -4xy & \text{for } 0 \leq x\leq 1 \ \text{and}\ 0 \leq y \leq 1\\ 0& \text{otherwise}\end{cases}\]
- Find the marginal densities of \(X\) and \(Y\)
- find \(f_{Y|X}( y \,|\, X=\frac14)\)
- 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.
- The diamond with vertices at \( (0,2), (-2,0), (0,-2), (2,0) \).
- 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 \(c>0\).
- Find \( c \).
- Find \(\mathbf{E}(X)\).
- Find \(\mathrm{Var}(X) \).
Infinite Mean
Suppose that \(X\) is a random variable whose density is
\[f(x)=\frac{1}{2(1+|x|)^2} \quad x \in (-\infty,\infty)\]
- Draw a graph of \(f(x)\).
- Find \(\mathbf{P}(-1 <X<2)\).
- Find \(\mathbf{P}(X>1)\).
- 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
\( f(x,y) = K(x^2+y^2), \quad 20 \leq x,y \leq 30 \)
- What is \(K\) ?
- Are the random variables independent ?
- What is the probability that both tires are underfilled ?
- What is the probability that \( |X-Y| \leq 3 \) ?
- What are the marginal densities ?
