Problem 3 - Basiswissen Statistik: Kompaktkurs für Anwender aus Wirtschaft, Informatik und Technik

Question

Geben Sie ein Beispiel an für ein Zufallsexperiment, bei dem unendlich viele Ausgänge vorkommen. (Geben Sie \(\Omega\) und \(P\) explizit an!)

Step-by-Step Solution

Answer

Observe the decay time of a radioactive atom with \(\Omega = [0, \infty)\) and \(P(T \leq t) = 1 - e^{-\lambda t}\), \(t \geq 0\).

Step 1: Understanding the Problem

We need to provide an example of a random experiment with infinitely many outcomes, along with specifying the sample space \( \Omega \) and the probability measure \( P \).

Step 2: Choosing the Random Experiment

Consider the experiment of observing the time until a radioactive atom decays. This is a common example where the outcomes are infinitely many, as time is continuous.

Step 3: Defining the Sample Space \(\Omega\)

The sample space \( \Omega \) includes all positive real numbers \([0, \infty)\), since decay time can be any non-negative real number.

Step 4: Specifying the Probability Measure \(P\)

The probability measure \( P \) for this experiment follows an exponential distribution, often used to model the time until a radioactive decay event. The probability density function is given by \( f(t) = \lambda e^{-\lambda t} \) for \( t \geq 0 \), where \( \lambda \) is the rate constant.

Step 5: Summary of the Experiment

Thus, we have defined an experiment of observing the decay time of a radioactive atom, with \( \Omega = [0, \infty) \) and the probability measure described by the exponential distribution \( P(T \leq t) = 1 - e^{-\lambda t} \), where \( T \) is the decay time.

Key Concepts

Probability Measure

In probability theory, a probability measure is a way to assign a likelihood to different events within our sample space. A probability measure helps us understand how likely it is for different outcomes to occur in a random experiment. This is particularly important in experiments with continuous outcomes, like the decay time of a radioactive atom.

A probability measure must satisfy certain conditions:

• Non-negativity: The probability of any event should not be negative.
• Normalization: The probability of the full sample space (all possible outcomes) sums up to 1.
• Additivity: If two events are disjoint, the probability of their union is the sum of their probabilities.

In our example, the probability measure is defined using the exponential distribution. This density function, given by \( f(t) = \lambda e^{-\lambda t} \), meets the aforementioned conditions and effectively models the decay event.

Sample Space

The sample space refers to all possible outcomes of a random experiment. It is essentially the universe of all potential results that we can observe.

When dealing with continuous random variables, the sample space can be quite large, even infinite. In our specific example of radioactive decay, the sample space \( \Omega \) consists of all positive real numbers. This is because decay time can occur at any fractional or whole number instant starting from zero and extending towards infinity.

This boundless range \( \Omega = [0, \infty) \) represents every possible moment where decay could occur, allowing us to measure the probability of decay happening by a certain time.

Exponential Distribution

The exponential distribution is a continuous probability distribution that is often used to model time until a specific event occurs, such as radioactive decay.

It is characterized by its probability density function \( f(t) = \lambda e^{-\lambda t} \), where \( \lambda \) is a positive constant known as the rate parameter. This parameter \( \lambda \) determines how "quickly" the decay happens – larger values of \( \lambda \) mean the event occurs more frequently or rapidly.

The key properties of the exponential distribution include:

• Memoryless: The probability of an event occurring in the future is independent of past events.
• Continuous: It applies to all real numbers within the sample space.

This distribution is integral to our problem as it defines the probability that the decay happens within a certain time frame \( P(T \leq t) = 1 - e^{-\lambda t} \).

Continuous Outcomes

Random experiments can have either discrete or continuous outcomes. Continuous outcomes allow for any value within a range, unlike discrete outcomes, which are fixed and countable.

In the context of our experiment - the time a radioactive atom takes to decay is an example of a continuous outcome. These outcomes can take on infinitely many values, such as fractions or irrational numbers, as they are not limited to integer values.

• Continuous outcomes mean higher precision in measurement.
• They require specific distributions to model and measure their probabilities, such as the exponential distribution.

Using a distribution like this helps us understand probabilities associated with any tiny slice of the timeline, giving us nuanced insight into when events actually occur.