Problem 3 - 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 \(X\). $$\begin{array}{lccccc}\hline \boldsymbol{x} & -2 & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & 1 / 16 & 4 / 16 & 6 / 16 & 4 / 16 & 1 / 16 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The mean (μ) of the random variable X is 0, the variance (σ^2) is \(\frac{3}{4}\), and the standard deviation (σ) is \(\frac{\sqrt{3}}{2}\).

Step 1: Compute the Mean (\(μ\))

First, we need to find the mean (μ) using the given probability distribution. We will use the formula \(μ = \sum_{i=1}^{n} x_i P(X=x_i)\). Mean (μ) = \((-2) * \frac{1}{16}\) + \((-1) * \frac{4}{16}\) + \(0 * \frac{6}{16}\) + \(1 * \frac{4}{16}\) + \(2 * \frac{1}{16}\) Mean (μ) = \(-\frac{1}{8} - \frac{1}{4} + \frac{1}{4} + \frac{1}{8}\) Mean (μ) = \(0\)

Step 2: Compute the Variance (\(σ^2\))

Next, we need to find the variance (σ^2) using the formula \( \sigma^2 = \sum_{i=1}^{n} (x_i - \mu)^2 P(X=x_i)\). Since we already know the mean (μ), we can plug it into the formula. Variance (σ^2) = \(((-2) - 0)^2 * \frac{1}{16}\) + \(((-1) - 0)^2 * \frac{4}{16}\) + \((0 - 0)^2 * \frac{6}{16}\) + \((1 - 0)^2 * \frac{4}{16}\) + \((2 - 0)^2 * \frac{1}{16}\) Variance (σ^2) = \(4 * \frac{1}{16} + 1 * \frac{4}{16} + 0 * \frac{6}{16} + 1 * \frac{4}{16} + 4 * \frac{1}{16}\) Variance (σ^2) = \(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\) Variance (σ^2) = \(\frac{3}{4}\)

Step 3: Compute the Standard Deviation (\(σ\))

Finally, we need to find the standard deviation (σ) which is the square root of the variance (σ^2). We already have the variance, so we just need to compute the square root. Standard deviation (σ) = \( \sqrt{\frac{3}{4}}\) Standard deviation (σ) = \(\frac{\sqrt{3}}{2}\)

Step 4: Present the Mean, Variance, and Standard Deviation

We have now computed the mean (μ), variance (σ^2), and standard deviation (σ) of the random variable X given the probability distribution. The results are: Mean (μ) = 0 Variance (σ^2) = \(\frac{3}{4}\) Standard deviation (σ) = \(\frac{\sqrt{3}}{2}\)

Key Concepts

Mean

The mean of a probability distribution is essentially the "average" or the expected value of a random variable. It gives you a sense of the central tendency of the data in the distribution. To calculate the mean, denoted by \( \, \mu \), for a discrete random variable like \( X \), you need to multiply each possible outcome \( x_i \) by its probability \( P(X = x_i) \) and then add all these products together. Consider our exercise with the outcomes: -2, -1, 0, 1, 2 and their respective probabilities. We calculate the mean as follows:

• Multiply each value of \( x \) by its corresponding probability.
• Add up all these values.

For example: \((-2) \times \frac{1}{16} + (-1) \times \frac{4}{16} + 0 \times \frac{6}{16} + 1 \times \frac{4}{16} + 2 \times \frac{1}{16}\). Applying the calculation, the mean of our distribution turns out to be \( 0 \). This tells us that, on average, the outcome of the random variable \( X \) is centered around 0.

Variance

Variance measures how spread out the values of a random variable are around the mean. If the variance is high, it means the numbers are spread out over a large range. If it's low, they are closer to the mean. For our discrete random variable, the variance \( \, \sigma^2 \, \) is calculated using the formula: \[ \sigma^2 = \sum_{i=1}^{n} (x_i - \mu)^2 \times P(X = x_i) \] Here is what you do:

• Subtract the mean from each \( x_i \) to find \( (x_i - \mu) \).
• Square each of these results to get rid of negative values, \( (x_i - \mu)^2 \).
• Multiply each squared result by its respective probability.
• Add all these products to get the variance.

In our example, performing these operations yields a variance of \( \frac{3}{4} \). This calculation indicates the average squared deviation from the mean.

Standard Deviation

The standard deviation is derived from the variance and provides a measure of dispersion around the mean that is in the same units as the random variable itself. To find the standard deviation \( \sigma \), you simply take the square root of the variance. In our case, the variance is \( \frac{3}{4} \), and thus the standard deviation is calculated as follows:\[ \sigma = \sqrt{ \frac{3}{4} } \] This gives us \( \frac{\sqrt{3}}{2} \), which is approximately 0.866. The standard deviation offers a clear insight into the variability of the distribution. A smaller standard deviation means the data points tend to be close to the mean, while a larger standard deviation indicates they are spread out. This concept is crucial when interpreting data distribution and probability in real-life scenarios, helping us understand how much the outcomes are expected to vary.