Question

Let \(X\) equal the duration (in min) of a telephone call that is received between midnight and noon and reported. The following times were reported 3.2, \(0.4,1.8,0.2,2.8,1.9,2.7,0.8,1.1,0.5,1.9,2,0.5,2.8,1.2,1.5,0.7,1.5,2.8,1.2\) Draw a probability histogram for the exponential distribution and a relative frequency histogram of the data on the same graph.

Short answer

The exponential distribution and the relative frequency histogram of the given phone call durations data are plotted on the same graph to provide a visual comparison. The lambda parameter of the exponential distribution is calculated as the reciprocal of the mean duration of the phone calls.

Step-by-step solution

Calculating lambda parameter

The parameter lambda for the Exponential distribution is derived as the reciprocal of the mean of the data. First, calculate the mean duration of the phone calls. Sum up all the reported times and then divide by the total number of reported times. The mean (mu) is given as: \[ mu = \frac{\sum_{i=1}^{n}X_i}{n} \]. After calculating the mean, compute lambda as: \[ lambda = \frac{1}{mu} \].

Plotting the exponential distribution

Now that the lambda parameter is known, the Exponential distribution can be plotted. Use the probability density function (pdf) of the Exponential distribution for plotting. The pdf of an Exponential distribution is given as: \[ f(x; lambda) = lambda * e^{-lambda * x} \] for \( x >= 0 \). Plot this function for the range of \( x \) that corresponds to the durations of the telephone calls.

Plotting the relative frequency histogram

Next, plot a histogram of the reported times. The heights of the bars of the histogram should correspond to the relative frequencies of the data instead of the absolute frequency. The relative frequency of a value is the absolute frequency of that value divided by the total number of data points.

Combining the plots

Now, combine the Exponential distribution plot from Step 2 and the relative frequency histogram from Step 3 onto the same graph. Use two different colors or styles of lines/bars to distinguish between the Exponential distribution and the histogram of the data. This will provide a visual comparison of how well the Exponential distribution fits the actual data.

Key concepts

Probability Histogram

A probability histogram is a graphical representation used for visualizing the probability distribution of a continuous random variable. In this context, it's particularly useful for illustrating the nature of an exponential distribution.
The exponential distribution is characterized by a single parameter, known as lambda, which describes the rate of occurrences. In a probability histogram, the heights of the bars represent the probability of the occurrence of each interval of values. This makes it easier to see the relation between different outcomes, providing a clear picture of the likelihood of various durations of phone calls between midnight and noon.
Creating a probability histogram involves calculating the probability of each range and plotting it. For exponential distributions, which model the time until an event, like a phone call, the probability histogram provides insight into what durations are most likely. Remember to label the axes clearly and choose appropriate intervals to ensure proper representation.

Relative Frequency Histogram

A relative frequency histogram is another important tool for data analysis. Instead of showing probabilities, this histogram illustrates how often each interval of values appears in a dataset compared to the total number of observations.
For the given telephone call durations, you would calculate the relative frequency by dividing the number of occurrences of a certain duration by the total number of calls. This gives a proportion that indicates how common different call durations are in the dataset.
When plotting the relative frequency histogram, bars are drawn with heights corresponding to these relative frequencies. It's a helpful way to normalize data, allowing comparison across datasets of different sizes. Combining this with the probability histogram on the same graph lets you visually assess how well the theoretical exponential distribution fits the actual data.

Lambda Parameter

The lambda parameter is a key component of the exponential distribution. It represents the rate at which events occur. A higher lambda value indicates that events happen quickly and frequently, while a lower value means they occur less frequently.
To find lambda, you need to calculate the mean () of the data, which is simply the sum of all reported times divided by the total number of reports. Once the mean is obtained, is the reciprocal: = \[ \frac{1}{\mu} \].
With lambda calculated, it can be plugged into the probability density function, \( f(x; \lambda) = \lambda \cdot e^{-\lambda \cdot x} \), to describe the exponential distribution. This formula gives you the likelihood of different durations for a phone call and is used to generate the probability histogram, helping visualize the exponential decay pattern typical of these scenarios.