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In probability, a linear transformation involves changing a random variable by applying linear operations, such as scaling or translation. Consider the random variable \( X \) with a mean \( \mu_X \) and standard deviation \( \sigma_X \). A linear transformation can be expressed mathematically as \( Y = aX + b \), where \( a \) and \( b \) are constants. In simpler terms:

• \( a \) is the scaling factor.
• \( b \) is the shift or translation factor.

For example, if \( X \) represents profits, \( a \) could be the percentage retained, and \( b \) could be a fixed deduction. In the provided exercise, the transformation formula is \( Y = 0.9X - 0.2 \), where:

• \( 0.9 \) scales the profit by 90%, meaning the firm retains 90% of the profits.
• \( -0.2 \) subtracts a fixed amount due to additional fees.

This transformation helps understand how external factors like fees affect overall profit.

The expected value is a key concept in probability, representing the average outcome one would expect from repeated trials of a random experiment. For a random variable \( X \) with discrete outcomes \( x_i \) and probabilities \( P(x_i) \), the expected value \( \mu_X \) is calculated by summing the products of each outcome and its probability:\[\mu_X = \sum x_i P(x_i)\]This expected value tells us the "center" or "balance point" of the distribution. In other words, if we were to repeat the investment scenario many times, the average profits would converge to the expected value.After applying a linear transformation to a random variable, the expected value also changes linearly. When \( Y = aX + b \), the expected value of \( Y \), denoted by \( \mu_Y \), is given by:\[\mu_Y = a\mu_X + b\]In the exercise, with \( \mu_X = 3 \), we find \( \mu_Y = 0.9 \times 3 - 0.2 = 2.5 \). This indicates that, after transformations, the average amount the firm expects to retain is 2.5 million dollars.

The standard deviation is a measure of the spread or variability of a set of values, providing insight into how much individual outcomes deviate from the expected value. For a random variable \( X \), the standard deviation \( \sigma_X \) quantifies the amount of variation or dispersion in the data.After a linear transformation of a random variable \( X \) into \( Y = aX + b \), the new standard deviation \( \sigma_Y \) is affected by the scaling factor \( a \). It is calculated as follows:\[\sigma_Y = |a| \sigma_X\]The absolute value is taken because the direction (up or down) of scaling does not impact the distance or variability measure—just the magnitude is relevant.For the investment problem, with the standard deviation of \( X \) being \( 2.52 \), the standard deviation of \( Y \) can be calculated as \( \sigma_Y = 0.9 \times 2.52 = 2.268 \). This tells us about the variability of the retained profit after accounting for consistent fees and deductions. A lower standard deviation suggests less uncertainty in the retained profits compared to the initial profits.