The following holds marginal pdf of X P(X > 3) = re-#(1+y) dy dx = xe-". e¬#" dy dx e¬ª dx = 0.05.
The following holds marginal pdf of X P(X > 3) = re-#(1+y) dy dx = xe-". e¬#" dy dx e¬ª dx = 0.05.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please explain the calculus portion of the attached screen shot without assuming too much. Thank you.
![In this educational excerpt, the process of computing the probability by integrating a joint probability density function (pdf) is discussed. The text explains two methods to compute the probability: either compute the marginal pdf of \( X \) first and then find \( X > 3 \) using the marginal pdf, or compute it directly. The focus here is on the second method, which is indirect.
The integral equation is presented as:
\[
P[(X,Y) \in A] = \int_A \int f(x,y) \, dx \, dy
\]
In this context, for a set \( A \), the above equation holds.
The main focus is on calculating \( P(X > 3) \) given by:
\[
P(X > 3) = \int_3^{\infty} \left( \int_0^{\infty} xe^{-x(1+y)} \, dy \right) \, dx = \int_3^{\infty} \left( \int_0^{\infty} xe^{-x} \cdot e^{-xy} \, dy \right) \, dx
\]
This simplifies through a series of steps involving integration:
1. First, the inner integral with respect to \( y \) is solved.
2. This yields a factor involving \( e^{-x} \).
3. The remaining integration over \( x \) results in evaluating an exponential function's limit.
The final result of the computation is:
\[
P(X > 3) = e^{-3} = 0.05
\]
This result denotes that the probability of \( X \) being greater than 3 is 0.05.
The explanation includes boxed content highlighting key information from the calculation, emphasizing the final probability value of 0.05.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f9cc468-5045-4ba5-8445-556d55f6f682%2F7afc3f6d-f730-4a75-90ca-83ed6641befb%2F98277wm_processed.png&w=3840&q=75)
Transcribed Image Text:In this educational excerpt, the process of computing the probability by integrating a joint probability density function (pdf) is discussed. The text explains two methods to compute the probability: either compute the marginal pdf of \( X \) first and then find \( X > 3 \) using the marginal pdf, or compute it directly. The focus here is on the second method, which is indirect.
The integral equation is presented as:
\[
P[(X,Y) \in A] = \int_A \int f(x,y) \, dx \, dy
\]
In this context, for a set \( A \), the above equation holds.
The main focus is on calculating \( P(X > 3) \) given by:
\[
P(X > 3) = \int_3^{\infty} \left( \int_0^{\infty} xe^{-x(1+y)} \, dy \right) \, dx = \int_3^{\infty} \left( \int_0^{\infty} xe^{-x} \cdot e^{-xy} \, dy \right) \, dx
\]
This simplifies through a series of steps involving integration:
1. First, the inner integral with respect to \( y \) is solved.
2. This yields a factor involving \( e^{-x} \).
3. The remaining integration over \( x \) results in evaluating an exponential function's limit.
The final result of the computation is:
\[
P(X > 3) = e^{-3} = 0.05
\]
This result denotes that the probability of \( X \) being greater than 3 is 0.05.
The explanation includes boxed content highlighting key information from the calculation, emphasizing the final probability value of 0.05.
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