The opening prices per share Y1 and Y2 of two similar stocks...

The opening prices per share $Y_{1}$ and $Y_{2}$ of two similar stocks are independent random variables, =ach with a density function given by $$f(y)=\left\{\begin{array}{ll} (1 / 2) e^{-(1 / 2)(y-4)}, & y \geq 4 \\ 0, & \text { elsewhere } \end{array}\right.$$ On a given morning, an investor is going to buy shares of whichever stock is less expensive. Find the a. probability density function for the price per share that the investor will pay. b. expected cost per share that the investor will pay.

The opening prices per share $Y_{1}$ and $Y_{2}$ of two similar stocks are independent random variables, =ach with a density function given by $$f(y)=\left\{\begin{array}{ll}
(1 / 2) e^{-(1 / 2)(y-4)}, & y \geq 4 \\
0, & \text { elsewhere }
\end{array}\right.$$ On a given morning, an investor is going to buy shares of whichever stock is less expensive. Find the
a. probability density function for the price per share that the investor will pay. b. expected cost per share that the investor will pay.

Key Concepts

Order Statistics

Order statistics concerns the properties and distributions of sorted random variables from a given sample. When choosing the minimum (or maximum) of several independent random variables, techniques from order statistics are used to derive the distribution of this value, making it a key concept for selecting the less expensive stock.

Transformation of Random Variables

This concept involves finding the probability distribution of a function of one or more random variables. In the context of this problem, the cost is a function of two independent stock prices, and the distribution of the minimum of these prices is derived by transforming the original distributions.

Expected Value Calculation

Expected value calculation is fundamental in probability theory as it represents the average outcome of a random variable. Here, it involves integrating the product of the determined probability density function and the value over its support to compute the investor's expected cost per share.

Exponential Distribution (Shifted)

The exponential distribution is widely used to model time until an event and is characterized by its rate parameter. In many problems, a shifted exponential distribution is used, meaning the standard exponential model is moved along the horizontal axis. Understanding its properties, such as its memoryless property and methods for integration, is crucial for handling problems involving such distributions.

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