2. Consider a gas station's daily sales. Let Y be the volume of gas sold per day in 1000s of gallons. Assume the distribution of Y follows this pdf: 1sxs2 S (v)=. else a. What value of k is required for the above function to be a valid pdf? b. Verify that the function given is a valid probability density function c. Determine the cumulative distribution function (cdf) of the random variable d. Compute the probability that there is exactly 1500 gallons of sales on a given day (Hint: FindP(r - 1.5) )
2. Consider a gas station's daily sales. Let Y be the volume of gas sold per day in 1000s of gallons. Assume the distribution of Y follows this pdf: 1sxs2 S (v)=. else a. What value of k is required for the above function to be a valid pdf? b. Verify that the function given is a valid probability density function c. Determine the cumulative distribution function (cdf) of the random variable d. Compute the probability that there is exactly 1500 gallons of sales on a given day (Hint: FindP(r - 1.5) )
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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![### Problem 2: Gas Station Daily Sales
Consider a gas station’s daily sales. Let \( Y \) be the volume of gas sold per day in **1000s of gallons**. Assume the distribution of \( Y \) follows this probability density function (pdf):
\[
f(y) =
\begin{cases}
k \left( y - \frac{1}{y^3} \right) & \text{for } 1 \leq y \leq 2 \\
0 & \text{elsewhere}
\end{cases}
\]
#### Questions:
a. **What value of \( k \) is required for the above function to be a valid pdf?**
b. **Verify that the function given is a valid probability density function.**
c. **Determine the cumulative distribution function (cdf) of the random variable.**
d. **Compute the probability that there is exactly 1500 gallons of sales on a given day.**
(Hint: Find \( P(Y = 1.5) \))](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b0a5411-bdb0-4804-8919-e2df25fb5bff%2Fe1d54587-3ac0-4df5-9544-9f9c17647711%2Fc22jja_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 2: Gas Station Daily Sales
Consider a gas station’s daily sales. Let \( Y \) be the volume of gas sold per day in **1000s of gallons**. Assume the distribution of \( Y \) follows this probability density function (pdf):
\[
f(y) =
\begin{cases}
k \left( y - \frac{1}{y^3} \right) & \text{for } 1 \leq y \leq 2 \\
0 & \text{elsewhere}
\end{cases}
\]
#### Questions:
a. **What value of \( k \) is required for the above function to be a valid pdf?**
b. **Verify that the function given is a valid probability density function.**
c. **Determine the cumulative distribution function (cdf) of the random variable.**
d. **Compute the probability that there is exactly 1500 gallons of sales on a given day.**
(Hint: Find \( P(Y = 1.5) \))
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