The lifespan X (years) of a species of pig has pdf f (x) = 3x2e−x^3 , 0 < x < ∞. What is the probability that a pig will live at least 1 year? Given that it lives 1 year, what is the conditional probability that it will live for at least another year?

The lifespan X (years) of a species of pig has pdf f (x) = 3x2e−x^3 , 0 < x < ∞. What is the probability that a pig will live at least 1 year? Given that it lives 1 year, what is the conditional probability that it will live for at least another year?

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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Question

The lifespan X (years) of a species of pig has pdf
f (x) = 3x²e^−x^{^3}, 0 < x < ∞.
What is the probability that a pig will live at least 1 year? Given that it lives 1 year, what is the conditional
probability that it will live for at least another year?

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Expert Solution

Step 1: Step by step solution for part a>

To find the probability that a pig will live at least 1 year, you need to calculate the integral of the probability density function (pdf) from 1 to infinity. The probability that a pig will live at least 1 year is given by:

$integral subscript 1 superscript infinity$

To solve this integral, you can make a substitution. Let $u = x^3$ , which implies $d u = 3 x^^{2} d x$ . The integral becomes:

$integral subscript 1 superscript infinity e x p left parenthesis negative u right parenthesis space d u$

Now, evaluate this integral from 1 to infinity:

$P (X \geq 1) =1 - exp(-1)$

So, the probability that a pig will live at least 1 year is $1 - exp(^{-1)}$ .

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