Variance Calculation A Comprehensive Solution For Random Variables

In probability theory and statistics, understanding the characteristics of random variables is crucial for making informed decisions and predictions. Among these characteristics, the mean (or expected value) and the variance play pivotal roles. This article delves into a specific problem, Q.75, which involves calculating the variance of a random variable and then applying this knowledge to find the variance of a transformed variable. We will break down the problem step by step, providing a clear and comprehensive explanation that caters to both beginners and advanced learners in statistics.

The problem at hand presents a discrete random variable X with a given probability distribution. The goal is to first calculate E[(X - E(X))^2], which represents the variance of X, and then to determine V(3X - 4), the variance of a linear transformation of X. This involves understanding the definitions of expected value and variance, as well as the properties of variance under linear transformations. Let's embark on this statistical journey to unravel the solution.

Understanding the Probability Distribution

The question provides the probability distribution of a discrete random variable X. A discrete random variable is a variable whose value can only take on a finite number of values or a countably infinite number of values. In this case, X can take on the values 4, 5, 7, 8, and 10. Each value has an associated probability, which indicates the likelihood of the random variable taking on that particular value. The probability distribution is given as follows:

This table tells us that:

• The probability of X being 4 is 0.15.
• The probability of X being 5 is 0.20.
• The probability of X being 7 is 0.40.
• The probability of X being 8 is 0.15.
• The probability of X being 10 is 0.10.

These probabilities sum up to 1, which is a fundamental property of any probability distribution. Now that we understand the distribution, we can proceed to calculate the expected value and variance.

Calculating the Expected Value E(X)

The expected value of a discrete random variable, denoted as E(X), is a measure of the central tendency of the variable. It is essentially the weighted average of the possible values of the variable, where the weights are the corresponding probabilities. The formula for the expected value of a discrete random variable is:

E(X) = Σ [x * P(x)]

where the summation is taken over all possible values of x. In our case, the expected value of X can be calculated as follows:

E(X) = (4 * 0.15) + (5 * 0.20) + (7 * 0.40) + (8 * 0.15) + (10 * 0.10)

Let's break down the calculation:

• 4 * 0.15 = 0.60
• 5 * 0.20 = 1.00
• 7 * 0.40 = 2.80
• 8 * 0.15 = 1.20
• 10 * 0.10 = 1.00

Adding these values together, we get:

E(X) = 0.60 + 1.00 + 2.80 + 1.20 + 1.00 = 6.60

Therefore, the expected value of the random variable X is 6.60. This value represents the average value we would expect X to take over many repetitions of the experiment.

Calculating the Variance E[(X - E(X))^2]

The variance of a random variable, denoted as Var(X) or E[(X - E(X))^2], measures the spread or dispersion of the variable's values around its mean (expected value). A higher variance indicates that the values are more spread out, while a lower variance indicates that they are clustered closer to the mean. The formula for the variance of a discrete random variable is:

Var(X) = E[(X - E(X))^2] = Σ [(x - E(X))^2 * P(x)]

where the summation is taken over all possible values of x. We already know that E(X) = 6.60. Now, we need to calculate (x - E(X))^2 for each value of X and then multiply it by the corresponding probability.

1. For X = 4: (4 - 6.60)^2 = (-2.60)^2 = 6.76 (6.76 * 0.15) = 1.014
2. For X = 5: (5 - 6.60)^2 = (-1.60)^2 = 2.56 (2.56 * 0.20) = 0.512
3. For X = 7: (7 - 6.60)^2 = (0.40)^2 = 0.16 (0.16 * 0.40) = 0.064
4. For X = 8: (8 - 6.60)^2 = (1.40)^2 = 1.96 (1.96 * 0.15) = 0.294
5. For X = 10: (10 - 6.60)^2 = (3.40)^2 = 11.56 (11.56 * 0.10) = 1.156

Now, we sum these values to find the variance:

Var(X) = 1.014 + 0.512 + 0.064 + 0.294 + 1.156 = 3.04

Therefore, the variance of the random variable X is 3.04. This value indicates the extent to which the values of X are spread out around the mean.

Calculating the Variance of a Linear Transformation V(3X - 4)

Now that we have calculated the variance of X, we need to find the variance of a linear transformation of X, specifically V(3X - 4). A linear transformation is a transformation of the form aX + b, where a and b are constants. In this case, a = 3 and b = -4. Understanding how variance behaves under linear transformations is crucial.

The key property we need to use is:

V(aX + b) = a^2 * V(X)

This property states that the variance of a linear transformation aX + b is equal to the square of the constant a multiplied by the variance of X. The constant b does not affect the variance because it simply shifts the distribution without changing its spread.

In our case, we have V(3X - 4). Using the property above, we get:

V(3X - 4) = 3^2 * V(X) = 9 * V(X)

We already calculated V(X) = 3.04. Therefore,

V(3X - 4) = 9 * 3.04 = 27.36

So, the variance of the transformed variable 3X - 4 is 27.36. This value is significantly higher than the variance of X because the multiplication by 3 has amplified the spread of the distribution.

Conclusion

In summary, we have successfully calculated the variance of the random variable X as 3.04 and the variance of the transformed variable 3X - 4 as 27.36. This problem demonstrates the importance of understanding the concepts of expected value and variance, as well as the properties of variance under linear transformations. These concepts are fundamental in statistics and probability theory, with applications in various fields such as finance, engineering, and data science.

Therefore, the correct answer is (b) 3.04, 27.36. This comprehensive solution not only provides the numerical answers but also elucidates the underlying principles and steps involved in calculating variances. By understanding these concepts, you can confidently tackle similar problems and apply statistical reasoning in real-world scenarios. Remember, the key to mastering statistics is not just memorizing formulas but understanding the logic and intuition behind them.

By walking through each step—from understanding the probability distribution to applying the linear transformation property—we've shown how to solve this problem and similar questions effectively. This detailed approach enhances not only problem-solving skills but also a deeper understanding of statistical concepts. This comprehensive exploration should solidify your understanding and enhance your problem-solving capabilities in statistics.