1. Uniform Random Variable: Model the actual resistive value R0 of a resistor marked as 100 ? with a 5% tolerance as a uniform random variable R0 over the range [95, 105] ?.
(a) Write the PDF of R0, fR0 (R0). Also sketch the PDF.
(b) What is the expected value of R0, E[R0]?
(c) What is the variance of R0, Var[R0]?
(d) What is the probability that the resistor value R0 falls within 1% of 100 ?, that is, between 99 to 101 ?? In other words, what is P(99 < R0 ≤ 101)?
2. Gaussian Random Variable: Model the lifetime of a laptop battery (maximum time it will run on battery power alone before needing recharging) as a Gaussian random variable X, with mean µx = 10 hours, and standard deviation σx = 2 hours. Use the tables for the CDF and Q-function of a standard normal random variable (posted on BbLearn in the Handouts folder) as needed.
(a) What is the probability that your laptop battery lifetime is less than or equal to 6 hours, P(X ≤ 6)?
(b) What is the probability that your laptop battery lifetime is between 8 and 12 hours, P(8 < X ≤ 12)?
(c) What is the probability that your laptop battery lifetime is greater than 18 hours, P(X > 18)? Give an actual number; don't just approximate the number as 0.
3. Exponential Random Variable, Joint PDF and Conditional PDF: You call two of your company's main clients regularly. The time in minutes of each call to client 1 is modeled as an exponential random variable T1 with λ = 1 phone call/20 minutes. The time in minutes of each call to client 2 is also modeled as an exponential random variable T2 with the same λ = 1 phone call/20 minutes. T1 and T2 are i.i.d. random variables; they are independent and identically distributed.
(a) Write an expression for the PDF of T1, fT1 (t1).
(b) What is the probability that a phone call to client 1 lasts longer than 20 minutes?
(c) Write an expression for the joint PDF fT1,T2 (t1, t2).
(d) Find the conditional PDF of T1 conditioned on T2, fT1|T2 (t1|t2).
From your conditional PDF, what is the probability that a phone call to client 1 lasts longer than 20 minutes, given that a phone call to client 2 lasted longer than 20 minutes?