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The Probability Mass Function (PMF) is an essential concept when dealing with discrete random variables. For a random variable, the PMF gives the probability that the random variable takes on a specific value. It's crucial when working with cumulative distribution functions (CDFs) because it helps derive specific probabilities.When you read a CDF, it provides the cumulative probability up to a certain point. To find the probability of an exact value, such as \(P(X=2)\), you use the PMF by subtracting the CDF value just before from the value at that point:- Example: \(P(X=2) = F(3) - F(2) = 0.39 - 0.19 = 0.20\)In summary, the PMF helps drill down into the specific probabilities from the cumulative probabilities presented in CDFs. Therefore, mastering the use of the PMF is indispensable for solving discrete probability problems.

A piecewise function is a function that is defined by different expressions over different intervals of its input. In probability, piecewise functions often represent the cumulative distribution function (CDF), which gives us the cumulative probabilities across defined intervals.For example, in our given CDF:- For \(x < 0\), \(F(x) = 0\)- For \(0 \leq x < 1\), \(F(x) = 0.06\)- And so on...Each segment or 'piece' of this function defines the probability data in its range, allowing us to understand how probability accumulates across intervals.Piecewise functions are incredibly useful in probability calculations as they give precise probability metrics at different phases or checkpoints of the CDF, providing a clear breakdown of how probability aggregates through the variable's domain.

The Complement Rule is a fundamental principle in probability theory used to simplify probability calculations. It states that the probability of an event not occurring is equal to one minus the probability of the event occurring. This rule is particularly useful when the direct calculation of a probability seems complex or when it is easier to calculate the probability of the opposite event. For example, to find \(P(X>3)\):- Instead of summing probabilities directly, use the complement of \(X \leq 3\):- So, \(P(X>3) = 1 - F(3) = 1 - 0.67 = 0.33\)The Complement Rule is not only a shortcut but also a crucial strategy for probability calculations, simplifying the computation process and helping you quickly find probabilities associated with the complementary events.

Probability Calculations are all about efficiently using mathematical tools to determine the likelihood of various outcomes. In our exercise, we had to calculate probabilities based on a CDF, such as \(P(2 \leq X \leq 5)\) and \(P(2 < X < 5)\).Steps usually involve:1. **Understanding the Question**: Break down what the probability asked for represents. Is it for exact values or a range?2. **Utilize CDF Data**: Use cumulative probabilities from a CDF. For ranges like \(P(2 \leq X \leq 5)\), subtract the CDF values at interval endpoints: - \(P(2 \leq X \leq 5) = F(5) - F(2) = 0.92 - 0.19 = 0.73\)3. **Adjust for Open Intervals**: For \(P(2 < X < 5)\), only adjust for endpoints: - \(P(2 < X < 5) = F(5) - F(3) = 0.92 - 0.39 = 0.53\)Following a systematic approach leads to clear, accurate probability calculations and ensures you account for all conditions specified in a probability problem.