f(x) = {o otherwise (a) Find the value of c which makes this a valid probability density function. (b) What is P(0 ≤ X ≤ ¹)? (c) Find the expected value and variance of X.
f(x) = {o otherwise (a) Find the value of c which makes this a valid probability density function. (b) What is P(0 ≤ X ≤ ¹)? (c) Find the expected value and variance of X.
A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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Please help with a-d: Let X be a continuous random variable which has a probability density f(x) given by:
![**Probability Density Function Problem**
Given the probability density function:
\[
f(x) =
\begin{cases}
cx^3 & \text{for } 0 \leq x \leq 1 \\
0 & \text{otherwise}
\end{cases}
\]
Answer the following questions:
(a) Find the value of \( c \) which makes this a valid probability density function.
(b) What is \( P(0 \leq X \leq \frac{1}{4}) \)?
(c) Find the expected value and variance of \( X \).
(d) Find the median of \( X \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3b47e7c-f6ee-4dde-9ef1-0f8b8a2d2829%2F5249f2f2-355e-4ace-a495-2dbbc3b89f5c%2Fltnaifa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Probability Density Function Problem**
Given the probability density function:
\[
f(x) =
\begin{cases}
cx^3 & \text{for } 0 \leq x \leq 1 \\
0 & \text{otherwise}
\end{cases}
\]
Answer the following questions:
(a) Find the value of \( c \) which makes this a valid probability density function.
(b) What is \( P(0 \leq X \leq \frac{1}{4}) \)?
(c) Find the expected value and variance of \( X \).
(d) Find the median of \( X \).
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