Consider the following:

Question

You will call ma back within next 20 minutes. The probability distribution is uniform, that means that there are no periods of time when your call is more likely (for example, the probability that you call during first 2 minutes is the same that you call during last 2 minutes and so on).

This is an example of a random variable with uniform continuous distribution.

To find probabilities of random variables with continuous distributions we will use so called the probability density functions.

Let's see how it works using the example above.

The graph of its density function is shown below. Find the area of the rectangle
. The are of the rectangle is equal to the probability of all possible values, in this case that you will call back within next 20 minutes. thus is why the area of the rectangle is
.

Answer 1

In a uniform continuous distribution, the probability density function (PDF) is constant over a specified interval. In your example, you will call back within the next 20 minutes. This means that the time until you call back can be considered a continuous random variable \(X\) that follows a uniform distribution over the interval \([0, 20]\) minutes.

The probability density function \(f(x)\) for a uniform distribution over the interval \([a, b]\) is defined as:

\[ f(x) = \frac{1}{b - a} \]

for \(a \leq x \leq b\), and \(f(x) = 0\) otherwise.

In this case, \(a = 0\) and \(b = 20\). Therefore, the PDF is:

\[ f(x) = \frac{1}{20 - 0} = \frac{1}{20} \]

for \(0 \leq x \leq 20\). This means that the height of the rectangle representing the PDF is \(\frac{1}{20}\).

The area of the rectangle is given by the width times the height. The width of the rectangle is the length of the interval, which is \(20 - 0 = 20\) minutes, and the height is \(\frac{1}{20}\).

Thus, the area of the rectangle is:

\[ \text{Area} = \text{width} \times \text{height} = 20 \times \frac{1}{20} = 1 \]

This area of 1 represents the total probability of all possible values, which in this case means that you will definitely call back at some time within the next 20 minutes. Therefore, the probability that you will call back within that time frame is:

\[ P(0 \leq X \leq 20) = 1 \]

In summary, the area of the rectangle representing the PDF of your calling time uniformly distributed over 20 minutes is 1, representing that there's a 100% probability that you will call back within the next 20 minutes.