x<0 A continuous random variable X has a cdf given by F(x)={x Osx<1. [o a) Find the pdf. y21 Compute Psxs b) using the cdf. Compute Psxs) c) using the pdf.
x<0 A continuous random variable X has a cdf given by F(x)={x Osx<1. [o a) Find the pdf. y21 Compute Psxs b) using the cdf. Compute Psxs) c) using the pdf.
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...
Related questions
Question
![**Topic: Continuous Random Variables and Probability Distributions**
**Problem:**
A continuous random variable \( X \) has a cumulative distribution function (cdf) given by:
\[
F(x) =
\begin{cases}
0 & \text{for } x < 0 \\
x^2 & \text{for } 0 \le x < 1 \\
1 & \text{for } x \ge 1
\end{cases}
\]
**Questions:**
a) Find the probability density function (pdf).
b) Compute \( P\left(\frac{1}{2} \le X \le \frac{3}{4}\right) \) using the cdf.
c) Compute \( P\left(\frac{1}{2} \le X \le \frac{3}{4}\right) \) using the pdf.
**Explanation:**
- To find the pdf from the cdf, differentiate \( F(x) \) with respect to \( x \).
- For part b, evaluate the cdf at the boundaries and subtract: \( F\left(\frac{3}{4}\right) - F\left(\frac{1}{2}\right) \).
- For part c, integrate the pdf from \( \frac{1}{2} \) to \( \frac{3}{4} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67df2f2b-9148-4dd5-b0ae-87328374df62%2F2ec50aed-6298-4339-8c77-0e43a841b0b0%2Fmofska_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Continuous Random Variables and Probability Distributions**
**Problem:**
A continuous random variable \( X \) has a cumulative distribution function (cdf) given by:
\[
F(x) =
\begin{cases}
0 & \text{for } x < 0 \\
x^2 & \text{for } 0 \le x < 1 \\
1 & \text{for } x \ge 1
\end{cases}
\]
**Questions:**
a) Find the probability density function (pdf).
b) Compute \( P\left(\frac{1}{2} \le X \le \frac{3}{4}\right) \) using the cdf.
c) Compute \( P\left(\frac{1}{2} \le X \le \frac{3}{4}\right) \) using the pdf.
**Explanation:**
- To find the pdf from the cdf, differentiate \( F(x) \) with respect to \( x \).
- For part b, evaluate the cdf at the boundaries and subtract: \( F\left(\frac{3}{4}\right) - F\left(\frac{1}{2}\right) \).
- For part c, integrate the pdf from \( \frac{1}{2} \) to \( \frac{3}{4} \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.Similar questions
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON