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A Probability Density Function (PDF) is a core concept in the study of continuous random variables, especially in distributions like the uniform distribution. It represents the likelihood of any outcome within a given range and is used to define probabilities over continuous intervals.
In the case of a uniform distribution on the interval \(a, b\), every value within this range is equally likely. Hence, the PDF for a uniform distribution between \(a\) and \(b\) is given by:

• \( f(x) = \frac{1}{b-a} \) for \( a \leq x \leq b \)
• \( f(x) = 0 \) for \(x < a\) or \(x > b\)

This means that any outcome between \(0\) and \(25\) for our given problem is equally probable and has the same density value of \(\frac{1}{25}\).
In practice, PDFs do not give probabilities for specific points but rather for intervals, making them crucial for calculating probabilities for continuous random variables.

The Cumulative Distribution Function (CDF) is another critical concept that relates closely to the PDF. While the PDF tells us about densities, the CDF provides the probability that a random variable \(X\) is less than or equal to a particular value \(x\).
For a uniform distribution, the CDF smoothly accumulates these probabilities over the interval. It is given by:

• \( F(x) = \frac{x-a}{b-a} \) for \( a \leq x \leq b \)
• \( F(x) = 0 \) for \(x < a\)
• \( F(x) = 1 \) for \(x > b\)

In the example of our exercise, with \(a = 0\) and \(b = 25\), the CDF becomes \(F(x) = \frac{x}{25}\) for \(0 \leq x \leq 25\). This means if you wanted to find the probability of \(X\) being less than or equal to a particular time, you read the CDF value directly at that time. Understanding the CDF helps in predicting cumulative probabilities and understanding the distribution's behavior over the given interval.

Integration is a fundamental mathematical tool that helps us find probabilities for continuous distributions by determining the area under the curve of the Probability Density Function (PDF). In the context of our exercise on uniform distribution, we use integration to compute specific probabilities.
To find the probability that \(X\) falls within some range \([c, d]\), we calculate the integral of the PDF across that interval:

• \( P(c \leq X \leq d) = \int_{c}^{d} f(x) \, dx \)

For the step-by-step solution given, this meant:

• For \(P(10 \leq X \leq 20)\), solving \( \int_{10}^{20} \frac{1}{25} \, dx \) results in \(0.4\).
• For \(P(X \geq 10)\), integrating \( \int_{10}^{25} \frac{1}{25} \, dx \) gives \(0.6\).

These integrals provide a practical means of calculating continuous probabilities, bridging between abstract mathematical concepts and real-world applications.