91Ó°ÊÓ

The joint probability density function (PDF) provides a way to describe the likelihood of multiple random variables occurring simultaneously. In our problem, we are dealing with a system of transistors, each having a lifetime that is exponentially distributed with a parameter \( \lambda \). The joint PDF is essentially the product of the individual PDFs of these random variables.

For \( n \) transistors, each represented as \( X_1, X_2, \ldots, X_n \), the joint PDF when assuming independence is calculated by:

• Multiplying the individual PDF of each transistor. Each PDF is \( f(x_i) = \lambda e^{-\lambda x_i} \).
• This results in the joint expression \( f(x_1, x_2, \ldots, x_n) = f(x_1)f(x_2) \cdots f(x_n) \).

This explains the simultaneous probabilities of various life spans of the random sample of \( n \) transistors.

Simplifying the multiplication gives us:
\[ f(x_1, x_2, \ldots, x_n) = \lambda^n \cdot e^{-\lambda (x_1 + x_2 + \cdots + x_n)} \]
This equation combines all the individual lifetimes into a single expression that considers their collective behavior.

When we say that random variables are independent and identically distributed (often abbreviated as IID), we mean that each random variable in question behaves the same mathematically, but independently. This is a crucial assumption for simplifying many probability problems.

Independence here implies that the lifetime of one transistor does not affect the lifespan of another. Mathematically, this is significant because it means that the joint PDF is simply the product of the individual PDFs.

• **Independent:** Any one transistor's lifetime is unaffected by others.
• **Identically Distributed:** Each lifetime follows the same exponential distribution with the same parameter \( \lambda \).

The IID assumption allows us to generalize our solution to the joint PDF across all transistors involved, simplifying calculations and making it easier to predict more complex scenarios.

The exponential random variable is a fundamental concept in probability and statistics, especially useful in modeling the time between independent events like the failure of components. In this case, it is applied to the lifespans of transistors.

The exponential distribution is characterized by its rate parameter \( \lambda \), and its probability density function (PDF) is defined as:
\[ f(x) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0 \]
Here's why the exponential distribution is particularly interesting and useful:

• **Memoryless Property:** The probability of failure in the next instant is independent of how long it has already survived.
• **Ease of Calculation:** The simplicity of its PDF makes it mathematically tractable.

Understanding the behavior of exponential random variables is crucial for predicting and analyzing the life expectancy of electronic components in a statistical manner.