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...
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![**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 \( \frac{3}{4} \).
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