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A random variable is a concept that helps in working with uncertain or random phenomena. In simple terms, it's a variable whose value depends on the outcomes of a random event. For Tipper Corporation, the random variable \( x \) represents their profit or loss in the next year. The values \( -1.2 \), \( -0.7 \), \( 0.9 \), and \( 2.3 \) millions denote specific possible outcomes from this uncertain situation.
Understanding that random variables can take on any value from a set of possibilities, we express them with probabilities. These probabilities describe the likelihood of each outcome occurring:

• \( x = -1.2 \) million with probability \( 0.17 \),
• \( x = -0.7 \) million with probability \( 0.21 \),
• \( x = 0.9 \) million with probability \( 0.37 \),
• \( x = 2.3 \) million with probability \( 0.25 \).

Knowing the values and associated probabilities forms what is called a probability distribution, paving the way for deeper analysis.

The mean of a probability distribution, often called the expected value, gives a measure of the "center" of the distribution. It is computed by summing up the products of each possible outcome with its probability. This process is concisely captured in the formula:
\[\mu = \sum x * p(x)\]
For Tipper Corporation, the calculation follows:

• \(-1.2 \times 0.17 = -0.204\),
• \(-0.7 \times 0.21 = -0.147\),
• \(0.9 \times 0.37 = 0.333\),
• \(2.3 \times 0.25 = 0.575\).

Adding these together gives a mean profit of \(0.557\) million dollars. The mean suggests that if the company were to experience these probabilities repeatedly over many years, the average annual profit would settle around \(0.557\) million dollars.

The standard deviation in a probability distribution quantifies the amount of variation or spread in a set of values. It gives a sense of how much the values might deviate from the mean. To calculate it, we use:
\[\sigma = \sqrt{\sum (x - \mu)^2 * p(x)}\]
This formula involves calculating the squared differences between each value and the mean, weighting them by their probabilities, and then taking the square root.
For the exercise:

• \((-1.2 - 0.557)^2 \times 0.17 = 0.526\),
• \((-0.7 - 0.557)^2 \times 0.21 = 0.333\),
• \((0.9 - 0.557)^2 \times 0.37 = 0.043\),
• \((2.3 - 0.557)^2 \times 0.25 = 0.755\).

Adding up these values gives \(1.657\), and taking the square root results in a standard deviation of approximately \(1.287\) million dollars. This tells us that, although the mean profit is \(0.557\) million, real outcomes will vary around this figure by \(1.287\) million with each annual occurrence.