91Ó°ÊÓ

The mean, also known as the expected value, is a fundamental concept in statistics. It provides a measure of the central tendency of a set of numbers. To calculate the mean of a discrete random variable, use the formula: \[ E[X] = \sum_{i} x_{i}P(x_{i}) \] Here, it signifies the sum of each outcome multiplied by its probability.
Essentially, it represents the algebraic average, considering the likelihood of each event.

• Identify each value of the random variable.
• Multiply each value by its probability.
• Add up all these products.

In our example:
If given a probability function \(R(x)=0.2\) for \(x=0,1,2,3,4\), substitute these into the formula. This gives us: \[ E[X] = 0*0.2 + 1*0.2 + 2*0.2 + 3*0.2 + 4*0.2 = 2.0 \]
This result, 2.0, indicates the mean value of our probability distribution.

Standard deviation is a statistical measure that shows how much variation exists from the mean. It is a short measure of spread or dispersion within a set of data points.
More intuitively, it tells us to what extent the values differ from the mean.

• A smaller standard deviation indicates that the values are closely grouped around the mean.
• A larger standard deviation suggests that the values are more spread out.

To find the standard deviation, first determine the variance:
Take the square root of that variance, as shown in the formula: \[ SD = \sqrt{Var[X]} \]
In this exercise, we have: \[ SD = \sqrt{2.0} \approx 1.41 \]
So, a standard deviation of approximately 1.41 illustrates the average distance of the data points from the mean in our data set.

Variance is a statistical measurement that quantifies the extent of spread in a set of data. It's essential to understand the variance to assess how varied or consistent a data set is. The formula for calculating variance \(Var[X]\) is \[ Var[X] = E[X^2] - (E[X])^2 \] Here's how to approach it:

• First, compute the expected value of the squared values of the random variable (\(E[X^2]\)).
• Calculate this by summing the square of each outcome multiplied by its probability.
• Then, use the previously calculated mean (squared) to complete the formula.

In our exercise: \[ E[X^2] = 0^2*0.2 + 1^2*0.2 + 2^2*0.2 + 3^2*0.2 + 4^2*0.2 = 6.0 \] Subtract square of mean from this value: \[ Var[X] = 6.0 - 2.0^2 = 2.0 \]
Understanding variance is key in determining data consistency, revealing how much the data set deviates from its mean.