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In probability theory, a random variable is a crucial concept that represents a real-valued function defined on a sample space. Simply put, it's a way to assign numerical values to the different outcomes of a probabilistic event. Random variables can be either discrete, taking on a finite or countable set of values, or continuous, where they can assume an infinite number of possible values within a given range. In our example with the contractor, the random variable \(x\) represents the profits in millions from bidding on state projects. The values \(x\) can take are \(0\), \(2\), \(5\), and \(10\) million dollars, depending on which jobs, if any, the contractor wins.
Understanding that each of these outcomes has an associated probability is vital. By evaluating the probability distribution of a random variable, you can grasp the likelihood of each potential result. This involves aligning each possible value of \(x\) with its probability \(P(x)\), providing a comprehensive overview of the expected outcomes of the contractor's bidding efforts. This arrangement allows us to perform further analysis, such as calculating the mean and standard deviation.

The mean, often referred to as the expected value in probability distributions, is a fundamental concept serving as the weighted average of all possible values. This allows us to determine a central or expected outcome in probabilistic scenarios. To calculate the mean of a random variable \(x\), you multiply each outcome by its probability and then add these products together. This is expressed mathematically as follows: \[ μ = ∑ [x \, P(x)] \] For our contractor example, this means summing up \((0 \times 0.10 + 2 \times 0.45 + 5 \times 0.30 + 10 \times 0.15)\). By calculating this, one can find the average profit expected from the contractor’s bids considering all probabilities. This provides insight into how much profit, on average, the contractor might anticipate.
The expected value allows businesses to plan strategies and manage resources based on likely outcomes. It captures the idea of what can be expected as an average, thus assisting in making informed decisions.

Standard deviation is a statistical measure that quantifies the variation or spread of a set of values. In the context of probability distributions, it indicates how much the outcomes of the random variable differ from the mean. To calculate standard deviation, one must first determine the variance, which is the expected value of the squared deviations from the mean:\[ σ^2 = ∑ [(x - μ)^2 \, P(x)] \] After obtaining the variance, the standard deviation \(σ\) is simply the square root of that value. In our scenario with the contractor, this involves computing the variance from the outcome probabilities and the already calculated mean:\((x - μ)\). Doing so provides understanding into the degree of unpredictability or risk associated with the contractor's potential profits.
A lower standard deviation suggests profits are consistent with the average expectation, while a higher deviation points to greater variability. Thus, for decision-making, knowing the standard deviation helps the contractor gauge potential profit risks, enabling more effective planning for various financial outcomes.