Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable—X for the right tire and Y for the left tire, with joint pdff(x, y) = K(x 2 + y 2 ) 20 ≤ x ≤ 30, 20 ≤ y ≤ 30 0 otherwisea. What is the value of K?b. What is the probability that both tires are underfilled?c. What is the probability that the difference in air pressure between the two tires is at most 2 psi?d. Determine the (marginal) distribution of air pressure in the right tire alone.e. Are X and Y independent rv’s?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable—X for the right tire and Y for the left tire, with joint pdf
f(x, y) = K(x 2 + y 2 ) 20 ≤ x ≤ 30, 20 ≤ y ≤ 30 0 otherwise
a. What is the value of K?
b. What is the
c. What is the probability that the difference in air pressure between the two tires is at most 2 psi?
d. Determine the (marginal) distribution of air pressure in the right tire alone.
e. Are X and Y independent rv’s?
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