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In a uniform distribution, the probability density function (PDF) represents how likely a random variable is to take any specific value within an interval. In our case, this interval is from 10 to 20. The PDF for a uniform distribution is characterized by its constant value throughout the interval. It is determined by the formula \( \frac{1}{b-a} \), where \( b \) and \( a \) are the endpoints of our interval. Here, the interval width is \( b-a = 20-10 = 10 \), so the height of the PDF is \( \frac{1}{10} \). This means that any outcome between 10 and 20 is equally likely. The graph of the PDF is a horizontal line because of this uniform likelihood, starting from \( x=10 \) to \( x=20 \). It highlights the idea that no single point within the range has more importance than another. Such an understanding is foundational for calculating probabilities in uniform distributions.

The expected value, often referred to as the mean, provides a central value for the distribution. For a uniform distribution, the expected value is calculated using the formula \( \frac{a+b}{2} \). This formula averages the lower and upper bounds of our interval. In our situation, \( a \) is 10 and \( b \) is 20, so the expected value is \( \frac{10+20}{2} = 15 \). This value serves as a fair representation of the "center" of the distribution range. It's important to understand that expected value helps summarize the distribution and gives us insight into the point around which values are evenly distributed. If you were repeatedly sampling values from this distribution, the average of those values would trend towards 15.

Variance is a measure of how much the values of a random variable spread out from the expected value. For a uniform distribution, variance is given by the formula \( \frac{(b-a)^2}{12} \). It reflects the degree to which the values differ from the mean. For our interval from 10 to 20, the calculation is \( \frac{(20-10)^2}{12} = \frac{100}{12} = \frac{25}{3} \approx 8.33 \). A larger variance indicates that the values are more spread out around the expected value. Conversely, a smaller variance means the values are tightly clustered around the mean. This concept is crucial because it helps us understand the consistency and reliability of our measurements within the distribution.

Computing probabilities in a uniform distribution involves calculating the area under the probability density function over an interval of interest.
For instance, to find \( P(x < 15) \), determine the area under the PDF from 10 to 15: \( \frac{1}{10} \times (15-10) = 0.5 \). This indicates a 50% chance of selecting a value less than 15.
Similarly, for \( P(12 \leq x \leq 18) \), it reflects the area from 12 to 18, which is \( \frac{1}{10} \times (18-12) = 0.6 \). Thus, there is a 60% probability for values to fall within this range. Understanding this methodology is vital because it applies geometric principles to probability calculations: the area under the curve corresponds to the probability of falling within a specific range, a fundamental notion in probability theory.