The Joint Pdf of Pressures for Right (X) and Left

Essay

The joint pdf of pressures for right (X) and left (Y) front tires is given by $f ( x , y ) = \left\{ \begin{array} { c l } K \left( x ^ { 2 } + y ^ { 2 } \right) & 20 \leq x \leq 30,20 \leq y \leq 30 \\0 & \text { otherwise }\end{array} \right.$ .
a. Determine the conditional pdf of Y given that X = x and the conditional pdf of X given that Y = y if you are given $F _ { X } ( x ) = 10 \mathrm { kx } ^ { 2 } + .05 \text { for } 20 \leq x \leq 30 \text { and } F _ { Y } ( y ) = 10 \mathrm {~ky} ^ { 2 } + .05 \text { for } 20 \leq y \leq 30$
b. If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a pressure of at least 25 psi? Compare this to $P ( Y \geq 25 )$
c. If the pressure in the right tire is found to be 22 psi, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire?