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In the context of uniform distribution, the probability density function (PDF) is quite straightforward. It assigns equal likelihood to all outcomes within the specified interval. This means that each point in the range has the same chance of occurring. The formula for a uniform distribution PDF over an interval \([a, b]\) is given by \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). This is because the total area under the probability density curve must equal one, representing certainty that the outcome will fall somewhere within \([a, b]\).
For the problem provided, \(X\) is uniformly distributed between \(0\) and \(25\). Therefore, the PDF can be derived as \(f(x) = \frac{1}{25-0} = \frac{1}{25}\), indicating a constant probability across the entire range.

The cumulative distribution function (CDF) offers insight into the probability that a random variable is less than or equal to a certain value. For a uniform distribution, the CDF starts at \(0\) when \(x < a\) and progresses linearly to 1 as \(x\) approaches and exceeds \(b\).
The CDF for a uniform distribution is given by:

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

For \(X\) uniformly distributed on \[0, 25\], the CDF simplifies to \(F(x) = \frac{x}{25}\) for \(0 \leq x \leq 25\). This expression captures the growing accumulation of probability as \(x\) increases from \(0\) up to \(25\).

The expected value, often seen as the 'average' or mean value a random variable will take, is a central concept in probability theory. For a uniform distribution, the expected value of a variable \(X\) is located exactly at the midpoint of the interval \([a, b]\). The formula to find this is \(E(X) = \frac{a+b}{2}\).
In our example where \(a = 0\) and \(b = 25\), this becomes \(E(X) = \frac{0+25}{2} = 12.5\). This means, if you were to repeatedly measure the time it takes for the read/write head to locate a record, the average time would be about 12.5 milliseconds.

Standard deviation provides a measure of the amount of variation or dispersion in a set of values. For uniform distributions, this measure illustrates how spread out the values are around the expected value. The formula for standard deviation of a uniform distribution is given by \(\sigma_{X} = \frac{b-a}{\sqrt{12}}\).
Applying this to the problem where the interval is \([0, 25]\), we calculate the standard deviation as \(\sigma_{X} = \frac{25-0}{\sqrt{12}} \), approximately \( 7.2169\). This indicates that the times it takes for the read/write head to find a record deviate from the mean by about 7.22 milliseconds, on average.