Problem 1 - 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 \(X\). $$\begin{array}{lllll}\hline x & 1 & 2 & 3 & 4 \\\\\hline P(X=x) & .4 & .3 & .2 & .1 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The mean, variance, and standard deviation of the random variable \(X\) are 3.2, 2.44, and approximately 1.56, respectively.

Step 1: Compute the Mean

Using the formula for the mean of a discrete random variable, we have: \(\mu = \sum_{i=1}^{n} x_i P(X=x_i) = (1)(0.4) + (2)(0.3) + (3)(0.2) + (4)(0.1) = 1.6 + 0.6 + 0.6 + 0.4 = 3.2\) The mean of the random variable \(X\) is 3.2.

Step 2: Compute the Variance

Using the formula for the variance of a discrete random variable, we have: \(\sigma^2 = \sum_{i=1}^{n}(x_i - \mu)^2 P(X=x_i)= (1-3.2)^2(0.4) + (2-3.2)^2(0.3) + (3-3.2)^2(0.2) + (4-3.2)^2(0.1)\) Calculating the terms, we get: \(\sigma^2 = (2.2)^2(0.4) + (-1.2)^2(0.3) + (-0.2)^2(0.2) + (0.8)^2(0.1) = 1.936 + 0.432 + 0.008 + 0.064 = 2.44\) The variance of the random variable \(X\) is 2.44.

Step 3: Compute the Standard Deviation

Using the formula for the standard deviation of a discrete random variable, we have: \(\sigma = \sqrt{\sigma^2} = \sqrt{2.44} \approx 1.56\) The standard deviation of the random variable \(X\) is approximately 1.56. In conclusion, the mean, variance, and standard deviation of the random variable \(X\) are 3.2, 2.44, and 1.56, respectively.

Key Concepts

Mean Calculation

The mean of a probability distribution is a fundamental statistical concept, often referred to as the "average." It gives you a central value of a random variable by considering all possible outcomes and their associated probabilities. This value provides a quick insight into where most of the probability mass lies. The calculation involves a weighted sum of all possible values.To determine the mean, denoted by \( \mu \), for a discrete random variable, use the formula:

• \( \mu = \sum_{i=1}^{n} x_i P(X=x_i) \)

Here, \( x_i \) is each possible value of the random variable \( X \), and \( P(X=x_i) \) is the probability of \( X \) taking the value \( x_i \). By multiplying each possible value by its probability and summing these products, you find the mean.In our example, the mean is calculated by:

• Mean = \( (1)(0.4) + (2)(0.3) + (3)(0.2) + (4)(0.1) = 3.2 \)

This reflects the "weight" of each number based on how probable it is. A higher mean indicates that the more probable outcomes are higher numbers.

Variance Calculation

Variance is a measure of how much the values of a random variable spread from the mean. It quantifies the degree of variation or dispersion in the distribution. The larger the variance, the more spread out the values are from the mean, while a small variance indicates they are close.The formula for variance \( \sigma^2 \) of a probability distribution is:

• \( \sigma^2 = \sum_{i=1}^{n}(x_i - \mu)^2 P(X=x_i) \)

Here, each value's deviation from the mean \( (x_i - \mu) \) is squared to remove negative signs and give weight to larger deviations. These squared deviations are then multiplied by their respective probabilities and summed up, providing a comprehensive view of variability.In the given exercise, the variance is found by:

• Variance = \( (1-3.2)^2(0.4) + (2-3.2)^2(0.3) + (3-3.2)^2(0.2) + (4-3.2)^2(0.1) = 2.44 \)

This indicates a moderate spread of values about the mean in our probability distribution.

Standard Deviation

The standard deviation is a useful statistic that provides insights into the spread and dispersion of a set of data values. As a square root of the variance, it is always non-negative and is in the same unit as the data, making it easier to interpret compared to the variance.With the variance \( \sigma^2 \) calculated, the standard deviation \( \sigma \) is:

• \( \sigma = \sqrt{\sigma^2} \)

This calculation helps in depicting how far on average each value deviates from the mean. A smaller standard deviation means that most values are close to the mean, while a larger one indicates that values are more spread out.In the context of our random variable, we calculated:

• Standard Deviation = \( \sqrt{2.44} \approx 1.56 \)

This numerical value helps us understand that the typical deviation of the random variable's values from the mean is about 1.56 units. It can be viewed as a reflection of the uncertainty or volatility inherent in the data distribution.