A supplier of kerosene has a weekly demand Y possessing a...

A supplier of kerosene has a weekly demand $Y$ possessing a probability density function given by $$f(y)=\left\{\begin{array}{ll} y, & 0 \leq y \leq 1 \\ 1, & 1<y \leq 1.5 \\ 0, & \text { elsewhere } \end{array}\right.$$ with measurements in hundreds of gallons. (This problem was introduced in Exercise $4.13 .$ ) The supplier's profit is given by $U=10 Y-4$ a. Find the probability density function for $U$ b. Use the answer to part (a) to find $E(U)$. c. Find $E(U)$ by the methods of Chapter 4

A supplier of kerosene has a weekly demand $Y$ possessing a probability density function given by $$f(y)=\left\{\begin{array}{ll}
y, & 0 \leq y \leq 1 \\
1, & 1<y \leq 1.5 \\
0, & \text { elsewhere }
\end{array}\right.$$
with measurements in hundreds of gallons. (This problem was introduced in Exercise $4.13 .$ ) The supplier's profit is given by $U=10 Y-4$
a. Find the probability density function for $U$
b. Use the answer to part (a) to find $E(U)$.
c. Find $E(U)$ by the methods of Chapter 4

Key Concepts

Transformation of Random Variables

This concept involves finding the distribution of a new random variable that is defined as a function of an original random variable. When a transformation is applied to a random variable, one typically relates the probability densities by using the inverse of the transformation, which then allows the derivation of the new probability density function (pdf) for the transformed variable.

Change of Variables Technique and Jacobian

When the transformation is monotonic, the change of variables technique is used to calculate the new pdf by applying the Jacobian (the absolute derivative of the inverse function). This approach adjusts the original pdf to account for the scaling effect of the transformation, ensuring that the probabilities are correctly mapped between the original and transformed variables.

Expected Value Calculation

The expected value (or mean) of a random variable is a key measure in probability and statistics that represents the long-run average outcome. It can be computed directly by integrating the product of the variable and its pdf over the entire range, or indirectly by using properties such as the law of the unconscious statistician, which allows the calculation of expectations of functions of random variables without explicitly finding the transformed pdf.

Handling Piecewise Probability Density Functions

Piecewise probability density functions require special attention because the function’s form changes over different segments of the random variable's domain. When conducting operations such as integration or transformation, each interval must be treated separately, with careful evaluation of limits and conditions to ensure that the overall function satisfies the properties of a valid probability distribution.

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