Random Variables in Economic Models Quiz

To find the expected value E[X] of the discrete random variable X, we multiply each possible return by its probability and sum the results: E[X] = (5% * 0.3) + (10% * 0.5) + (-5% * 0.2) = 1.5 + 5 - 1 = 5.5%. However, the correct answer is 6.5%, indicating a possible miscalculation in the probabilities or returns provided.

A continuous random variable can take any value within a specified range or interval, allowing for an infinite number of possible outcomes. This characteristic is essential in economic models, where variables like price, income, or consumption can vary smoothly rather than being restricted to discrete values.

To find the variance Var(X), use the formula Var(X) = E[X²] - (E[X])². Here, E[X] is 10 and E[X²] is 150. Calculating, Var(X) = 150 - (10)² = 150 - 100 = 50. Thus, the variance of X is 50.

To find E[Y], we substitute E[X] into the profit function. Given Y = 20X - 50 and E[X] = 15, we calculate E[Y] as follows: E[Y] = 20 * E[X] - 50 = 20 * 15 - 50 = 300 - 50 = 250. Thus, the expected profit is 250.

The normal distribution effectively represents a wide range of real-world phenomena due to the central limit theorem, which states that the sum of many independent random variables tends to be normally distributed. This makes it a valuable tool in economics for modeling variables like income, stock prices, and other factors that exhibit randomness and variability.

When two random variables X and Y are independent, the occurrence of one does not affect the occurrence of the other. This lack of influence means that their covariance, which measures the degree to which they vary together, is zero. Thus, Cov(X, Y) = 0 indicates no linear relationship between the two variables.

The standard deviation of a random variable is the square root of its variance. Given that the variance of W is 25, the standard deviation is calculated as √25, which equals 5. This value indicates the extent of variation or dispersion of the wealth variable around its mean.

In portfolio theory, the variance of a two-asset portfolio is influenced not only by the individual variances of the assets but also by their covariance. Covariance measures how the returns of the two assets move together, affecting the overall risk of the portfolio. A positive covariance increases risk, while a negative one can reduce it.

Linearity of expectation refers to the principle that the expected value of a linear transformation of a random variable can be expressed as a linear combination of constants and the expected value of the variable. This property simplifies calculations involving expected values, making it a fundamental concept in probability and statistics.

For a uniform distribution on the interval [a, b], the expected value E[X] is calculated using the formula E[X] = (a + b) / 2. Here, a = 0 and b = 10, so E[X] = (0 + 10) / 2 = 5.0, representing the midpoint of the distribution range.

Using a random variable to model demand enables economists to quantify the uncertainty surrounding demand fluctuations. This approach helps in assessing the likelihood of various outcomes, allowing firms to evaluate potential risks associated with production, inventory management, and pricing strategies, ultimately aiding in more informed decision-making under uncertainty.

The cumulative distribution function (CDF) F(x) provides the probability that a random variable X takes a value less than or equal to x. It applies to both discrete random variables, where probabilities are assigned to specific outcomes, and continuous random variables, where probabilities are derived from areas under the curve of the probability density function.