Question

Berechnen Sie die Mittelwerte der Zufallsvariablen \(x\), die nach den folgenden Normalverteilungen verteilt sind a) \(f(x)=\frac{1}{3 \sqrt{2 \pi}} e^{-\frac{(x-2)^{2}}{18}}\) b) \(f(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{(x+4)^{2}}{2}}\)

Short answer

a) 2b) -4

Step-by-step solution

Identify the mean (Mittelwert) in a Normal Distribution

For a normal distribution given by the function \ f(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{(x-\mu)^2}{2 \sigma^2}}, \ \ the mean \( \mu \) is the parameter that represents the location of the center of the distribution.

Extract the mean for part a

Given: \ f(x)= \frac{1}{3 \sqrt{2 \pi}} e^{- \frac{(x-2)^2}{18}} This equation matches the form of a normal distribution. In this case, \ \mu = 2 Thus, the mean of the random variable \(x\) in part a is \( 2 \).

Extract the mean for part b

Given: \ f(x)= \frac{1}{\sqrt{2 \pi}} e^{- \frac{(x+4)^2}{2}} This equation also matches the form of a normal distribution. Here, \ \mu = -4 Thus, the mean of the random variable \(x\) in part b is \( -4 \).

Key concepts

Zufallsvariable (Random Variable)

A Zufallsvariable, or random variable, is a key concept in probability and statistics. It represents a numerical outcome of a random phenomenon. In simpler terms, it's a variable whose values are determined by the results of a random process.

Imagine rolling a fair six-sided die. The random variable, in this case, could be the number that appears on the die face. So, the Zufallsvariable might be noted as \( X \) and could take any value from 1 to 6. The probability of each outcome can be assigned based on how likely it is to be rolled.

There are two types of random variables: discrete and continuous. Discrete Zufallsvariable take on distinct, separate values (like the die example), while continuous Zufallsvariable can take any value within a range (such as measuring the height of individuals). Understanding random variables in the scope of Normalverteilung (normal distribution) is crucial because many statistical methods and theories are built around them.

Mittelwert (Mean)

The Mittelwert, or mean, of a random variable is the average of all possible values it can take, weighted by their respective probabilities. If you think of the center of a normal distribution curve, that's the mean.

In a Normalverteilung (normal distribution), the mean is represented by the Greek letter \( \mu \). This parameter tells us where the center of the distribution is located. For example, in the function \( f(x)= \frac{1}{3 \sqrt{2 \pi}} e^{-\frac{(x-2)^{2}}{18}} \), \( \mu \) is 2, indicating that the center of this normal distribution is at the value 2.

The mean is significant because it provides a measure of the central location of the data. It is also a key component in the calculation and understanding of other statistical measures, such as variance and standard deviation. If you remember nothing else, remember that the mean is essentially the balancing point of the probability distribution.

Parameter (Parameter)

In the context of a normal distribution, a parameter is a value that defines certain characteristics of the distribution. For the normal distribution function \( f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}} \), there are two main parameters: \( \mu \) (mean) and \( \sigma \) (standard deviation).

The parameter \( \mu \) specifies the location of the peak of the distribution, essentially where the center lies. The parameter \( \sigma \) describes the spread or 'width' of the distribution. If \( \sigma \) is small, the curve is narrow and tall, indicating that the data points are closely clustered around the mean. If \( \sigma \) is large, the curve is wider and shorter, showing more spread out data.

These parameters are critical in understanding the shape and spread of the normal distribution. By knowing \( \mu \) and \( \sigma \), you can completely describe the distribution's behavior. This is why it's important to be able to identify these parameters from the normal distribution function, as shown in the examples of the original exercise.