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

question 6

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?

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Definitions:

Pulsation

A rhythmic throbbing or beating, often related to the heartbeat or the flow of blood within the arteries.

Foreign Bodies

Objects or substances found within the body that are not naturally produced by it, which can sometimes lead to infection or injury.

Sterile Gauze

A medical dressing made from a thin, transparent fabric that has been sterilized to prevent infection when applied to wounds.

Human Bite

An injury resulting from a person biting another person, potentially leading to infection and other complications.