The ages of a group of executives attending a convention are...

The ages of a group of executives attending a convention are uniformly distributed between 35 and 65 years. If the random variable $X$ denotes ages in years, the probability density function is as follows: $$ f(x)= \begin{cases}1 / 30 & \text { for } 35<x<65 \\ 0 & \text { for all other values of } x\end{cases} $$ a. Graph the probability density function for $X$. b. Find and graph the cumulative distribution function for $X$. c. Find the probability that the age of a randomly chosen executive in this group is between 40 and 50 years. d. Find the mean age of the executives in the group.

The ages of a group of executives attending a convention are uniformly distributed between 35 and 65 years. If the random variable $X$ denotes ages in years, the probability density function is as follows:
$$
f(x)= \begin{cases}1 / 30 & \text { for } 35<x<65 \\ 0 & \text { for all other values of } x\end{cases}
$$
a. Graph the probability density function for $X$.
b. Find and graph the cumulative distribution function for $X$.
c. Find the probability that the age of a randomly chosen executive in this group is between 40 and 50 years.
d. Find the mean age of the executives in the group.

Key Concepts

Uniform Distribution

Uniform distribution is a probability distribution where all outcomes in a given interval are equally likely. In continuous cases, this distribution is characterized by a constant probability density within the interval, and zero outside. Its simplicity makes it a foundational concept in probability theory and statistics for modeling events with equal likelihood over a fixed range.

Probability Density Function

The probability density function (pdf) for a continuous random variable is a function that describes the relative likelihood for the variable to take on a given value. In the context of a uniform distribution, the pdf is constant across the interval of possible outcomes. Graphically, it is represented as a horizontal line across the interval, reflecting equal probability for every subinterval of equal length.

Cumulative Distribution Function

The cumulative distribution function (cdf) of a continuous random variable gives the probability that the variable takes a value less than or equal to a specific value. The cdf is obtained by integrating the pdf from the lower bound of the distribution up to the value in question. For a uniform distribution, the cdf is a linear function within the defined interval, starting at 0 at the lower bound and reaching 1 at the upper bound.

Probability Calculation Over an Interval

Calculating the probability that a random variable falls within a particular interval involves integrating the pdf over that interval or, alternatively, computing the difference in the cdf values at the endpoints of the interval. This method is frequently used for determining the likelihood of an outcome occurring within a specific range, especially in cases of uniform distributions.

Expected Value/Mean of a Uniform Distribution

The expected value or mean of a uniform distribution on a given interval is calculated as the midpoint of that interval. This represents the central tendency of the distribution, indicating where outcomes are centered over the span of the possible values. The mean provides a summary statistic that is particularly simple to compute owing to the symmetry inherent in the uniform distribution.

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