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2. In commuting to work, a professor must first get on a bus near his house and then transfer to a second bus. If the waiting time (in minutes) at each stop has the density; 0 10 a. Sketch fx(x) b. Show that fx (x) is legitimate c. What is the probbility that total waiting time is at most 3 min? What is the probbility that total waiting time is at most 8 min ? d. e. What is the probbility that total waiting time is between 3 and 8 min? f. What is the probability that total waiting time is either less than 2 min or more than 6 min?

2. In commuting to work, a professor must first get on a bus near his house and then transfer to a second bus. If the waiting time (in minutes) at each stop has the density; 0 10 a. Sketch fx(x) b. Show that fx (x) is legitimate c. What is the probbility that total waiting time is at most 3 min? What is the probbility that total waiting time is at most 8 min ? d. e. What is the probbility that total waiting time is between 3 and 8 min? f. What is the probability that total waiting time is either less than 2 min or more than 6 min?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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1:43 ← MAY LTE il 34% & V Tesy Comp COT UB June 4, 7:52 AM College of Technology Probability and Statistics (EEC338) Tutorial 03 1. The lifetime of a bulb expressed in weeks is a Rayleigh random variable with probability density function given by: 2400 fx(x) = 200 x20 x < 0, (a) Show that fx (x) is legitimate. (b) Deduce the distribution function Fx (x). Hence or otherwise, (c) What is the probability that the bulb will not last a full week? What is the corresponding probability that the bulb lifetime will exceed a full week? 2. In commuting to work, a professor must first get on a bus near his house and then transfer to a second bus. If the waiting time (in minutes) at each stop has the density; 0≤x<5 25 fx(x)=- 5 ≤x≤ 10 25 0 x <0 or x> 10. a. Sketch fx(x) b. Show that fx (x) is legitimate c. What is the probbility that total waiting time is at most 3 min? d. What is the probbility that total waiting time is at most 8 min ? e. What is the probbility that total waiting time is between 3 and 8 min? f. What is the probability that total waiting time is either less than 2 min or more than 6 min? 3. The input-output relationship in a system is described by y = 16e-1,6x where x is the input and y is the output. The input is in the range of -1≤ x ≤ 2, and all values in this range occur in an equally likely fashion. (a) Draw the sample spaces of the input and the output (b) Find P(Y≤ 16) and P(2≤ Y <20) 4. Find the values of the real constants a and b such that Fx(x) = [1-a exp(-x/b)]u(x) is a valid distribution function. 5. A random variable X is Gaussian with #x= 0 and ox=1 (a) What is the probability that IX] > 2? (b) What is the probability that X > 2? 6. Repeat exercise 3 if x = 4 and ax=2 7. Let X be a random variable with density fx (x). Let another random variable Y be defined by Y=(X-a)². Find (y) in terms of Fx (x). Hence suppose X is a random Gaussian variable where 1 -x² fx(x) = exp( (2πσχέ 20x2 Find the probability density function Fy(y) Tutor: Takembo Clovis (L) ||| < O
Transcribed Image Text:1:43 ← MAY LTE il 34% & V Tesy Comp COT UB June 4, 7:52 AM College of Technology Probability and Statistics (EEC338) Tutorial 03 1. The lifetime of a bulb expressed in weeks is a Rayleigh random variable with probability density function given by: 2400 fx(x) = 200 x20 x < 0, (a) Show that fx (x) is legitimate. (b) Deduce the distribution function Fx (x). Hence or otherwise, (c) What is the probability that the bulb will not last a full week? What is the corresponding probability that the bulb lifetime will exceed a full week? 2. In commuting to work, a professor must first get on a bus near his house and then transfer to a second bus. If the waiting time (in minutes) at each stop has the density; 0≤x<5 25 fx(x)=- 5 ≤x≤ 10 25 0 x <0 or x> 10. a. Sketch fx(x) b. Show that fx (x) is legitimate c. What is the probbility that total waiting time is at most 3 min? d. What is the probbility that total waiting time is at most 8 min ? e. What is the probbility that total waiting time is between 3 and 8 min? f. What is the probability that total waiting time is either less than 2 min or more than 6 min? 3. The input-output relationship in a system is described by y = 16e-1,6x where x is the input and y is the output. The input is in the range of -1≤ x ≤ 2, and all values in this range occur in an equally likely fashion. (a) Draw the sample spaces of the input and the output (b) Find P(Y≤ 16) and P(2≤ Y <20) 4. Find the values of the real constants a and b such that Fx(x) = [1-a exp(-x/b)]u(x) is a valid distribution function. 5. A random variable X is Gaussian with #x= 0 and ox=1 (a) What is the probability that IX] > 2? (b) What is the probability that X > 2? 6. Repeat exercise 3 if x = 4 and ax=2 7. Let X be a random variable with density fx (x). Let another random variable Y be defined by Y=(X-a)². Find (y) in terms of Fx (x). Hence suppose X is a random Gaussian variable where 1 -x² fx(x) = exp( (2πσχέ 20x2 Find the probability density function Fy(y) Tutor: Takembo Clovis (L) ||| < O
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