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A piecewise function is essentially a function built from multiple sub-functions, each applying to a particular interval of the main functions domain. To better visualize a piecewise function, imagine stitching different linear functions together, where each `piece` or segment has its own rule.

• Each piece operates under its own condition or interval.
• The transition from one piece to another might lead to a "step" in the function's values.

In the given exercise, the function \( F(x) \) is a piecewise function that defines the Cumulative Distribution Function (CDF). This CDF is made up of several segments, each covering different intervals of \( x \): - From \(-\infty\) to \(-10\), the probability is 0.- From \(-10\) to 30, the probability is 0.25.- From 30 to 50, the probability increases to 0.75.- From 50 onwards, it reaches the maximum probability of 1.
Piecewise functions like \( F(x) \) are often used in statistics to represent situations where probabilities change in steps at certain thresholds. Understanding the transition and the flat segments is crucial for effectively determining probabilities in specified ranges.

Probability calculation considers the likelihood of an event within a specific interval. It's typically extracted using the definition of a probability distribution, often represented by a function.
In the context of \( F(x) \), a Cumulative Distribution Function (CDF):

• \( P(X \leq x) \) gives the probability that a random variable \( X \) is less than or equal to a value \( x \).
• To find the probability for any interval, assess the function values at interval boundaries.
• This could involve calculating differences to find the probability between two points.

For example, when determining \( P(40 \leq X \leq 60) \):

• You know \( P(X \leq 60) = 1 \) (since the function steps to 1 at \( x = 50 \))
• And \( P(X \leq 40) = 0.75 \)
• Thus, \( P(40 \leq X \leq 60) = 1 - 0.75 = 0.25 \).

Calculating these probabilities involves understanding where boundaries lie in the function, and how the function steps through these thresholds.

When dealing with a CDF expressed as a piecewise function, tackling each probability question can be clearer with a step-by-step approach.

• Step 1: Examine Intervals: Start by identifying which interval your query falls into. This will tell you the piecewise segment to use, as well as constant probabilities or steps associated with it.
• Step 2: Interpret CDF Value: Use the piecewise function to get the CDF values at relevant points within these intervals. Carefully observe where the function changes value.
• Step 3: Calculation Based on Boundaries: If calculating probability between boundaries, subtract the smaller CDF value at one boundary from the larger CDF value at the other. This difference gives you the probability for the interval considered.

Each step serves to break down what might seem complex into manageable parts, emphasizing comprehension over rote calculation. This thoughtful process helps ensure accurate interpretation and calculation of probabilities, vital for understanding distributions and predictions made from them.