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The Probability Density Function (PDF) for a uniform distribution helps to determine the likelihood of a random variable falling within a particular range. For a uniform distribution, every outcome in the interval is equally likely to occur. This means the graph of the PDF is a flat, horizontal line.

In a uniform distribution defined between two points, such as our example with bounds \(A = -5\) and \(B = 5\), the PDF is given by the formula \( f(x) = \frac{1}{B-A} \) for \( A \leq x \leq B \). For our parameters, this simplifies to a constant value of \(\frac{1}{10}\) across the interval.

This PDF can be used to compute probabilities by calculating the area under the curve. Since the curve is flat, the probability between any two points is simply the length of the interval divided by \(10\), which is the total length from \(A\) to \(B\).

Continuous Probability Distributions apply to continuous random variables, which can take any value within a given range. Unlike discrete distributions, which concern specific values, continuous distributions deal with intervals. In our problem, we consider the interval from \(-5\) to \(5\).

A uniform distribution is one of the simplest types of continuous probability distributions. It is characterized by a constant PDF, indicating that every value within this range is equally probable. This means the probability of the variable taking any single, specific value is literally zero; instead, we consider probabilities across intervals.

• The length of the interval directly determines the probability under a uniform distribution; larger intervals have higher probabilities.
• The total area under the PDF curve between \(A\) and \(B\) sums up to \(1\), representing the total certainty that the variable is somewhere within this range.

Understanding continuous distributions is foundational to solving probability tasks involving such variables.

Calculating probabilities in uniform distributions is straightforward since each outcome in the specified interval is equally likely. The key is to properly understand the range of interest and apply the basic formula for uniform probabilities.

In the given exercise, we compute probabilities by finding the proportion of the interval over the entire length from \(A\) to \(B\). For instance, to find \(P(X < 0)\), calculate the length from \(-5\) to \(0\) and divide by total interval length \(10\):
\[ P(X < 0) = \frac{0 - (-5)}{10} = 0.5 \]

Similarly, for intervals \(P(-2.5 < X < 2.5)\) and \(P(-2 \leq X \leq 3)\), the procedure remains the same:

• Find the length of the desired interval.
• Divide by the total interval length of \(10\).

For a flexible approach, like finding \(P(k < X < k+4)\), ensure the specified range \(k\) satisfies given conditions: \(-5 < k < k+4 < 5\). Then, apply the same method for probability, leading to a constant result for any \(k\) within this range, which is \(0.4\).

This methodology highlights the simplicity and elegance of uniform probability calculations.