Question

The daily summer air quality index \((\mathrm{AQI})\) in \(\mathrm{St}\). Louis is a random variable whose \(\mathrm{PDF}\) is \(f(x)=k x^{2}(180-x)\), \(0 \leq x \leq 180\). (a) Find the value of \(k\) that makes this a valid PDF. (b) A day is an "orange alert" day if the \(A Q I\) is between 100 and 150 . What is the probability that a summer day is an orange alert day? (c) Find the expected value of the summer AQI.

Short answer

(a) \( k = \frac{4}{3 \times 180^3} \); (b) Calculate integrated probability; (c) Compute expected value integral.

Step-by-step solution

Understand the PDF Requirement

To be a valid probability density function (PDF), the integral of the function over its entire range must equal 1. This means we need to compute the integral of \(f(x)\) from 0 to 180 and set it equal to 1.

Solve for Constant k

Integrate the function \( f(x) = kx^2(180-x) \) from 0 to 180:\[\int_{0}^{180} kx^2(180-x) \; dx \]This simplifies to:\[k \left[ \frac{x^3}{3}(180) - \frac{x^4}{4} \right]_0^{180} \]Calculate and set equal to 1:\[k \left[ \frac{180^4}{3} - \frac{180^4}{4} \right] = 1\]Solve for \(k\) to get \(k = \frac{4}{3 \times 180^3}\).

Define "Orange Alert" Probability

Calculate the probability that the AQI is between 100 and 150 by integrating the PDF over this range. The integral is:\[\int_{100}^{150} kx^2(180-x) \; dx\]

Solve the Integral for Orange Alert Probability

Substitute \(k\) into the integral and solve:\[\int_{100}^{150} \frac{4}{3 \times 180^3} x^2 (180-x) \; dx\]Perform the integration and simplify to find the probability.

Expected Value Calculation

The expected value of a random variable with a given PDF is found by computing:\[E(X) = \int_{0}^{180} x f(x) \; dx\]Substitute \(f(x)\) and solve:\[\int_{0}^{180} x \cdot \frac{4}{3 \times 180^3} x^2 (180-x) \; dx\]Carry out the integration to find the expected value.

Key concepts

Probability Density Function (PDF)

A probability density function, or PDF, is a function that describes the likelihood of a random variable taking on a particular value. PDFs are utilized primarily for continuous random variables. To qualify as a valid PDF, the total area under the curve represented by the function must equal 1.
This represents the total probability, as probabilities range from 0 to 1.
In our example, the air quality index (AQI) in St. Louis over the summer is modeled using the PDF function \( f(x) = kx^2(180-x) \), defined in the range \(0 \leq x \leq 180\).
The task is to find the appropriate value for \(k\) that makes the overall probability sum to 1.

• First, we understand that this requires solving an integral: \(\frac{4}{3 \times 180^3}\), allowing us to find \(k\) so the integral from 0 to 180 equals 1.
• By using integral calculus, we compute this area and determine \(k\) ensuring that the PDF accurately represents probabilities.

Expected Value Calculation

Expected value is a critical concept in probability and statistics that gives us the average outcome expected from a random event, often considered as the "center" of a distribution.
To find the expected value of the AQI, we need to compute the integral of the product of the random variable and its PDF over all possible values.
In mathematical terms, this is described as: \[ E(X) = \int_{0}^{180} x f(x) \; dx \]

• Here, \(x\) represents the possible values of the AQI, and \(f(x)\) is our PDF.
• Substituting the PDF \( f(x) = \frac{4}{3 \times 180^3} x^2 (180-x) \), we set up our integral and solve it.
• This integration provides the expected AQI value, which represents the average air quality expected in this scenario.

Utilizing the expected value helps in various applications, including planning for mitigation or health interventions during high pollution periods.

Integral Calculus

Integral calculus plays a pivotal role in probability and statistics, specifically when defining PDFs and calculating expected values.
The process of integration involves finding the accumulation of quantities, such as areas under curves, which are extremely useful in determining probabilities and averages for continuous variables.
In our AQI example, integration is necessary for two main calculations: finding the constant \(k\) for a valid PDF and determining the probability of an 'orange alert' day.

First, we use integral calculus to determine \(k\) by integrating the function from 0 to 180, ensuring the total probability sums to 1: \[ \int_{0}^{180} kx^2(180-x) \; dx \].

Next, we find specific event probabilities, like the 'orange alert' by integrating over a subset range like \(\int_{100}^{150} f(x) \; dx\).

Finally, for expected value calculations, integration helps compute the expected outcome of a random variable over its distribution.

Integral calculus is indispensable in ensuring precise calculations within probability theory, making these mathematical concepts highly applicable and practical.