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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 32EQ

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QUESTION 4
Suppose X is a random variable with probability density function ƒ₁ (x) and Y is a random
variable with density function ƒ₂ (x). Then X and Y are called independent random variables if
their joint density function is the product of their individual density functions:
ƒ(x, y) = f(x)ƒ₂ (y)
We modelled waiting times by using exponential density functions
if t < 0
ƒ(1)={u²³e-4₂
f(t)
if t≥0
where is the average waiting time. In the next example, we consider a situation with two
independent waiting times.
The joint density function of X and Y is a function f of two variables such that the probability
(x, y) lied in the region R is
P((x, y) = R) = ſ[ƒ (x,y) dA
[[
R
The manager of a bank determines that the average time customers wait in line to take a queue
number is 2 minutes and the average waiting time they queue before receiving the service at the
counter is 20 minutes.
Assuming that the waiting times are independent, find the probability that customers waits a total
of less than 30 minutes before their turn.
Hint: Find P(X+Y<30). Sketch the region R.

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