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In a uniform distribution, the probability density function (pdf) is what defines how the probability is spread across the interval. Unlike other distributions which may have more complex functions, the pdf in a uniform distribution is a simple horizontal line. It has the same height at every point in the interval because every outcome is equally likely.
For this exercise, the random variable \( X \) is uniformly distributed on the interval \([0, 25]\). The formula to find the pdf is given by \( f(x) = \frac{1}{b-a} \), where \(a\) and \(b\) are the bounds of the distribution.
Therefore, the pdf for this interval is \( f(x) = \frac{1}{25} \). This function tells us that any time between 0 and 25 milliseconds is equally probable, and the probability spread is even across all these points. You can think of the pdf as showing what share of an event's total probability happens at any given time within the range.

The Cumulative Distribution Function (CDF) of a random variable provides the probability that the variable will take a value less or equal to a specific value. In other words, it maps every point to the cumulative probability up to that point. For a uniform distribution, the CDF is a linear function due to its constant pdf.
The CDF for a uniform distribution over the interval \([a, b]\) is given by \( F(x) = \frac{x-a}{b-a} \) for \(a \leq x \leq b\). This formula gives us the cumulative probability up to any point \(x\) within the interval.
Specifically, for our interval \([0, 25]\), it simplifies to \( F(x) = \frac{x}{25} \). This means, if you select any time point \( x \) in milliseconds within this range, \( F(x) \) will tell you the probability that the chosen time is less than or equal to \( x \). For \( x < 0 \), \( F(x) = 0 \), and for \( x > 25 \), \( F(x) = 1 \), indicating that all probability is accounted for beyond these bounds.

The expected value, often denoted as \( E(X) \), of a random variable is the average or mean value that it takes on over a large number of trials. For a uniform distribution, this is the midpoint of the interval.
The formula to determine the expected value for a uniform distribution over the interval \([a, b]\) is \( E(X) = \frac{a+b}{2} \). This makes intuitive sense, as in a uniform distribution, all values are equally likely, and thus the mean falls right in the center.
Applying this formula to our interval \([0, 25]\), we find \( E(X) = \frac{0+25}{2} = 12.5 \). This tells you that, on average, the time it takes the read/write head to locate a record should be around 12.5 milliseconds.

Standard deviation is a measure of spread in a set of values. It tells us how much the values of a random variable deviate from the mean on average. In a uniform distribution, this quantifies how spread out the values are over the interval.
The formula for the standard deviation \( \sigma_X \) of a uniform distribution over the interval \([a, b]\) is \( \sigma_X = \sqrt{\frac{(b-a)^2}{12}} \). This formula calculates the average amount that values will differ from the expected value.
For the interval \([0, 25]\), the standard deviation is \( \sigma_X = \sqrt{\frac{(25-0)^2}{12}} = \sqrt{\frac{625}{12}} \approx 7.21 \). This tells you that times typically vary from the mean by about 7.21 milliseconds, giving you an idea of the variability in the times.