Problem Pi. Consider the following probability density function (pdf) for some continuous random variable X defined by { * 372 f (x) = 64 if 0 1) (b)The expected value of X, (i.e., E(X)) (c)The expected value of X², (i.e., E(X²) ) (d)The variance of X, (i.e., V(X) )
Problem Pi. Consider the following probability density function (pdf) for some continuous random variable X defined by { * 372 f (x) = 64 if 0 1) (b)The expected value of X, (i.e., E(X)) (c)The expected value of X², (i.e., E(X²) ) (d)The variance of X, (i.e., V(X) )
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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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F285c1a6b-569d-42a1-9c0c-69bff961af9e%2Fe5d78c2c-f7cd-4a53-a326-000a55bd1889%2Fbu9kyvn_processed.png&w=3840&q=75)
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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