Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The probability distribution of a random variable \(X\) is given. Compute the mean, variance, and standard deviation of \(X\). $$\begin{array}{lllllll}\hline \boldsymbol{x} & 10 & 11 & 12 & 13 & 14 & 15 \\\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & 1 / 8 & 2 / 8 & 1 / 8 & 2 / 8 & 1 / 8 & 1 / 8 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The mean (Expected Value) of the random variable \(X\) is \(E[X] = 12\). The expected value of the square of the random variable \(X\), i.e., \(E[X^2] = 146\). Using these values, the variance of the random variable \(X\) is \(Var(X) = 2\). Finally, the standard deviation of the random variable \(X\) is \(SD(X) = \sqrt{2}\).

Step 1: Compute the mean (Expected Value)

Calculate the mean of the random variable \(X\) by multiplying the values of \(x\) with their respective probabilities and summing up the products. Using the given probability distribution, the mean is computed as: \(E[X] = (10 \times \frac{1}{8}) + (11 \times \frac{2}{8}) + (12 \times \frac{1}{8}) + (13 \times \frac{2}{8}) + (14 \times \frac{1}{8}) + (15 \times \frac{1}{8})\)

Step 2: Compute the value of \(E[X^2]\)

Now, we need to calculate the expected value of the square of the random variable \(X\), i.e., \(E[X^2]\). This can be done using the same process as in step 1, but with the square of \(x\) instead: \(E[X^2] = (10^2 \times \frac{1}{8}) + (11^2 \times \frac{2}{8}) + (12^2 \times \frac{1}{8}) + (13^2 \times \frac{2}{8}) + (14^2 \times \frac{1}{8}) + (15^2 \times \frac{1}{8})\)

Step 3: Compute the Variance

With \(E[X]\) and \(E[X^2]\) calculated, we can now compute the variance using the formula mentioned in the analysis: \(Var(X) = E[X^2] - (E[X])^2\)

Step 4: Compute the Standard Deviation

Finally, the standard deviation can be calculated by taking the square root of the variance: \(SD(X) = \sqrt{Var(X)}\) After completing each step, you will have the mean, variance, and standard deviation of the random variable \(X\).

Key Concepts

Mean

The mean, also known as the expected value, is a measure of the central tendency of a probability distribution. It provides a single number that describes the average outcome you would expect if you repeated an experiment many times.
In probability distributions, calculating the mean involves multiplying each possible value (\(x\)) by its probability (\(P(X = x)\)) and then summing these products.

• Formula: \(E[X] = \sum (x \cdot P(X = x))\)

This formula tells us that for each value of \(x\), we find its contribution to the mean by weighing it according to how often it occurs. In our example, the values 10, 11, 12, 13, 14, and 15 are considered, each multiplied by their respective probabilities, and then added together to find \(E[X]\).
Thus, the mean is a powerful tool to give a quick snapshot of where the bulk of the probability mass is located in the distribution.

Variance

Variance is a measure that tells us how much the values of a random variable differ from the mean. It's a crucial concept for understanding the spread of the distribution.To calculate variance, you first determine the expected value of the squared deviations from the mean. This involves two main steps:

1. Calculate the expected value of the square of the random variable, \(E[X^2]\).

• Formula: \(E[X^2] = \sum (x^2 \cdot P(X = x))\)

2. Use the variance formula:

• \(Var(X) = E[X^2] - (E[X])^2\)

By squaring each value of the random variable before multiplying it by its probability, you give more weight to larger deviations. Once you subtract the square of the mean, you get the variance. This value can help you understand how spread out the values of the random variable are in relation to the mean.
Variance is always non-negative and a larger variance indicates more spread in the distribution.

Standard Deviation

Standard deviation is another measure of spread in a distribution, but it is more intuitive than variance. It indicates how much individual data points typically deviate from the mean.The standard deviation is simply the square root of the variance. This transformation brings the units of variance back to the same units as data, making it easier to interpret.

• Formula: \(SD(X) = \sqrt{Var(X)}\)

Understanding standard deviation is crucial because it gives a real-sense feeling of variation in the data set. In our example, once the variance is computed, taking its square root provides the standard deviation, offering a straightforward depiction of how data disperses from the average across the distribution.
Generally, a small standard deviation implies the values are close to the mean, while a large standard deviation signals a wide spread of values. This knowledge is instrumental for statisticians and researchers when analyzing data trends and variability.