Question

In Exercises \(\mathrm{P} .78\) to \(\mathrm{P} .81,\) use the probability function given in the table to calculate: (a) The mean of the random variable (b) The standard deviation of the random variable $$ \begin{array}{lccc} \hline x & 1 & 2 & 3 \\ \hline p(x) & 0.2 & 0.3 & 0.5 \\ \hline \end{array} $$

Short answer

The mean of the random variable is 2.3 and the standard deviation of the random variable is approximately 0.89.

Step-by-step solution

Calculate the Mean

To find the mean of the random variable 'x', multiply each value of 'x' by its respective probability and then sum up these values. Here that would be \( \mu = (1*0.2) + (2*0.3) + (3*0.5)\)

Calculation

Perform the calculations and get the mean of the random variable. \( \mu = 0.2 + 0.6 + 1.5 = 2.3\)

Calculate Standard Deviation

Use the calculated mean to find the standard deviation. Substitute the values in the formula, \( \sigma = \sqrt{\sum((x-\mu)^2*p(x))}\), which leads to \( \sigma = \sqrt{( (1-2.3)^2*0.2 + (2-2.3)^2*0.3 + (3-2.3)^2*0.5 )}\)

Calculation

Perform the calculations and get the standard deviation of the random variable. \( \sigma = \sqrt{0.52 + 0.027 + 0.245} = \sqrt{0.792}\)

Final Calculation of Standard Deviation

Calculate the square root to find the final value of the standard deviation. \( \sigma = \sqrt{0.792} = 0.89 \) (rounded to two decimal places)

Key concepts

Mean of Random Variable

The mean of a random variable, also known as the expected value, represents the average outcome we would expect to see if we were to repeat a random experiment many times. It's a crucial concept in probability and statistics as it provides a measure of the central tendency of the probable outcomes.

To calculate the mean \(\mu\), we take each possible value that the random variable \(x\) can take, multiply it by its corresponding probability \(p(x)\), and then sum up these products. In the context of our exercise:

\[\mu = (1 \cdot 0.2) + (2 \cdot 0.3) + (3 \cdot 0.5)\]

Doing the math gives us:

\[\mu = 0.2 + 0.6 + 1.5 = 2.3\]

The mean of the random variable in our example is 2.3. This provides an expected value, suggesting that if the random process behind the distribution were to be repeated, on average, the outcome would be close to 2.3.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

To calculate the standard deviation \(\sigma\) of a random variable, we use the mean \(\mu\) computed earlier. Each value of the random variable is subtracted from the mean, squared, multiplied by its probability, and these new values are summed up. Afterward, we take the square root of that sum to find the standard deviation.

Calculating Standard Deviation

\[\sigma = \sqrt{\sum ((x - \mu)^2 \cdot p(x))}\]

For our exercise, the substitution gives:

\[\sigma = \sqrt{((1 - 2.3)^2 \cdot 0.2) + ((2 - 2.3)^2 \cdot 0.3) + ((3 - 2.3)^2 \cdot 0.5)}\]

Putting the values through the formula:

\[\sigma = \sqrt{0.52 + 0.027 + 0.245} = \sqrt{0.792}\]

After calculating the square root:

\[\sigma = \sqrt{0.792} = 0.89 \]

Here, 0.89 is the standard deviation, rounded to two decimal places. It tells us the typical distance of the values from the mean within this probability distribution.

Random Variable

A random variable is a numerical description of the outcome of a statistical experiment. It assigns a unique numerical value to every potential outcome of a random process. Random variables can be discrete, taking on a finite or countably infinite set of values, or continuous, taking on any value in a given range.

In our exercise:

• A discrete random variable \(x\) is given with three possible outcomes: 1, 2, and 3.

• The associated probabilities for these outcomes are given by the probability function \(p(x)\), where \(p(1) = 0.2, p(2) = 0.3\), and \(p(3) = 0.5\).

This probability function tells us how likely each outcome is. For instance, in this scenario, we're more likely to observe the outcome '3' as it has the highest probability, 0.5.

Through these concepts, we can understand the behavior of random variables within the realm of probability, which is essential in many fields, including mathematics, engineering, economics, and more.