A First Course in Probability (10th Edition)
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ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Please solve P1
![**Problem P₁.**
Consider the following probability density function (pdf) for some continuous random variable \( X \) defined by
\[
f(x) =
\begin{cases}
\frac{3x^2}{64} & \text{if } 0 \leq x \leq 4 \\
0 & \text{elsewhere}
\end{cases}
\]
Compute each of the following:
(a) \( p(x \geq 1) \)
(b) The expected value of \( X \), (i.e., \( E(X) \) )
(c) The expected value of \( X^2 \), (i.e., \( E(X^2) \) )
(d) The variance of \( X \), (i.e., \( V(X) \) )
(e) Find the 85th percentile value of \( X \)? (Hint: Find \( M \) such that \( p(x \leq M) = .85 \) )
---
This problem requires evaluating several properties of a continuous random variable \( X \) using its probability density function over the specified range. The tasks involve calculating probabilities, expected values, and the variance, as well as determining a specific percentile value.](https://content.bartleby.com/qna-images/question/285c1a6b-569d-42a1-9c0c-69bff961af9e/e5d78c2c-f7cd-4a53-a326-000a55bd1889/bu9kyvn_processed.png)
Transcribed Image Text:**Problem P₁.**
Consider the following probability density function (pdf) for some continuous random variable \( X \) defined by
\[
f(x) =
\begin{cases}
\frac{3x^2}{64} & \text{if } 0 \leq x \leq 4 \\
0 & \text{elsewhere}
\end{cases}
\]
Compute each of the following:
(a) \( p(x \geq 1) \)
(b) The expected value of \( X \), (i.e., \( E(X) \) )
(c) The expected value of \( X^2 \), (i.e., \( E(X^2) \) )
(d) The variance of \( X \), (i.e., \( V(X) \) )
(e) Find the 85th percentile value of \( X \)? (Hint: Find \( M \) such that \( p(x \leq M) = .85 \) )
---
This problem requires evaluating several properties of a continuous random variable \( X \) using its probability density function over the specified range. The tasks involve calculating probabilities, expected values, and the variance, as well as determining a specific percentile value.
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Probability
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Author:Sheldon Ross
Publisher:PEARSON