Determining the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) for a Uniform Distribution | Math Insights

The time it takes for a college student to travel between her home and her college is uniformly distributed between 40 and 90 minutes .

To determine the probability density function (pdf) for this uniform distribution, we need to calculate the height of the pdf at each point within the given range

To determine the probability density function (pdf) for this uniform distribution, we need to calculate the height of the pdf at each point within the given range.

In a uniform distribution, the pdf is constant within the range and zero outside the range. The height of the pdf is determined by the total probability being equal to 1.

Given that the range of travel time is from 40 to 90 minutes, the total range is 90 – 40 = 50 minutes. The probability density over this range needs to be equal to 1. Therefore, the height of the pdf is 1/50.

Let’s denote the random variable representing the travel time as X, and the pdf as f(x). Then, for any given value of x within the range, the pdf can be defined as:

f(x) = 1/50, if 40 <= x <= 90 f(x) = 0, otherwise The cumulative distribution function (cdf), F(x), gives the probability that the travel time is less than or equal to x. In this case, the cdf will increase linearly from 0 to 1 over the range of 40 to 90 minutes. F(x) = 0, if x < 40 F(x) = (x - 40) / 50, if 40 <= x <= 90 F(x) = 1, if x > 90

This cdf can be interpreted as follows: if x is less than 40, the probability of the travel time being less than or equal to x is 0. If x is between 40 and 90, the probability increases linearly from 0 to 1 as x increases within that range. Finally, if x is greater than 90, the probability becomes 1.

These calculations provide a mathematical description of the given uniform distribution for the travel time of the college student between her home and college.

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