Answered: The lifespan X (years) of a species of…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Related questions

Question

The lifespan X (years) of a species of pig has pdf
f (x) = 3x²e^−x^{^3}, 0 < x < ∞.
Does X possess the memoryless property? Eplain your answer. Find the first quartile m0.25, median m0.5, and third quartile m0.75 for pigs of this species. What do each of these quantities represent?

Expert Solution

Step 1: Determine Random Variable X

To determine whether the random variable X, representing the lifespan of a species of pig, possesses the memoryless property, we need to check if it satisfies the following property:

Memoryless Property: For a random variable X, it has the memoryless property if, for all positive values a and b, the following equation holds:

$P left parenthesis X greater than space a space plus space b space vertical line space X greater than space a right parenthesis space equals space P left parenthesis X greater than space b right parenthesis$

In simpler terms, it means that the probability that the pig's lifespan exceeds (a + b), given that it has already survived to age a, should be the same as the probability that the pig's lifespan exceeds b. This property is often associated with exponential distributions.

Let's calculate this for the given probability density function (pdf):

$f left parenthesis x right parenthesis space equals space 3 x hat 2 space asterisk times space e hat left parenthesis negative x hat 3 right parenthesis comma space 0 space less than x infinity$

First, we need to calculate the conditional probability:

$P left parenthesis X space & g t semicolon space a space plus space b space vertical line space X space & g t semicolon space a right parenthesis space equals space integral left square bracket a space plus space b comma space infinity right square bracket space left parenthesis 3 x hat 2 space asterisk times space e hat left parenthesis negative x hat 3 right parenthesis right parenthesis space d x space divided by space integral left square bracket a comma space infinity right square bracket space left parenthesis 3 x hat 2 space asterisk times space e hat left parenthesis negative x hat 3 right parenthesis right parenthesis space d x$

Now, let's find the integral values and simplify:

$integral left square bracket a plus b comma infinity right square bracket left parenthesis 3 x squared asterisk times e to the power of left parenthesis minus x cubed right parenthesis right parenthesis d x equals integral left square bracket a plus b comma infinity right square bracket 3 x squared asterisk times e to the power of left parenthesis minus x cubed right parenthesis d x$
$integral left square bracket a comma infinity right square bracket left parenthesis 3 x squared asterisk times e to the power of left parenthesis minus x cubed right parenthesis right parenthesis d x equals integral left square bracket a comma infinity right square bracket 3 x squared asterisk times e to the power of left parenthesis minus x cubed right parenthesis d x$

Now, we need to calculate these integrals. Unfortunately, the integrals of this form do not have elementary solutions, and we would need to rely on numerical methods to compute these values.

However, for an exponential distribution, the memoryless property holds true, which implies that the species of pig in question does not have a memoryless property. Therefore, you will need to use numerical methods to calculate the conditional probabilities and demonstrate this.