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Chapter 23: Problem 1

Explain what is meant by a probability density function.

Short Answer

Expert verified
Answer: A probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value or occurring within a specific range of values. It is used to describe continuous probability distributions, where the outcomes can take infinitely many values within a given range. To compute the probability of a continuous random variable within a specified range, we calculate the integral of the PDF over that range. For example, if X is a continuous random variable with PDF f(x) and we want to find the probability that X lies within the range [c, d], we would compute: P(c 鈮� X 鈮� d) = 鈭玙{c}^{d} f(x) dx.

Step by step solution

01

Definition: Probability Density Function (PDF)

A probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value or occurring within a specific range of values. It does so by assigning probabilities to different intervals on the real line. The PDF is used to describe continuous probability distributions, where the outcomes can take infinitely many values within a given range.
02

Properties of Probability Density Functions

There are two main properties of a probability density function that must be satisfied: 1. The PDF must be non-negative for all values of the random variable: For a continuous random variable X and its PDF f(x), we must have f(x) 鈮� 0 for all x. This is because the probability of any event cannot be negative. 2. The integral of the PDF over the entire range of the random variable must equal 1: Since the PDF assigns probabilities to all possible values of the random variable, it must be the case that the sum of probabilities of all possible outcomes is 1. In mathematical terms, for a PDF f(x) and a random variable X with the range [a, b], the integral of f(x) from a to b must equal 1: 鈭玙{a}^{b} f(x) dx = 1
03

Interpreting a Probability Density Function

A probability density function does not give the probability of a specific value directly, but rather the probability for the value to be in a given interval. To find the probability of a continuous random variable being within a certain range, we calculate the integral of the PDF over that range. For example, if X is a continuous random variable with PDF f(x) and we want to find the probability that X lies within the range [c, d], we would compute: P(c 鈮� X 鈮� d) = 鈭玙{c}^{d} f(x) dx It is essential to note that the probability of a continuous random variable taking a specific value is zero. This is because there are infinitely many values that the random variable can take on, so the probability of it being equal to one specific value is infinitesimally small. In summary, a probability density function is a function that characterizes the distribution of a continuous random variable. It adheres to specific properties, and its integral is used to compute probabilities of the random variable within specified ranges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Random Variable
A continuous random variable is a type of random variable that takes on an infinite number of possible values within a given range. Unlike discrete random variables that have a countable number of outcomes, continuous random variables are characterized by a smooth and uninterrupted line of values. For example, the height of students in a classroom or the temperature on a certain day are continuous random variables because they can be measured with any level of precision, within a certain range.

Here are some key characteristics of continuous random variables:
  • They are represented by an interval on the number line. This means that a continuous range of possible outcomes exists.
  • Values of a continuous random variable cannot be listed individually due to the infinite possibilities, as they exist within a certain interval and are not physically countable.
  • A probability density function (PDF) is used to describe the probabilities associated with these variables, where the actual probabilities are found over intervals rather than at specific points.
Probability Distribution
A probability distribution is a mathematical description that characterizes and models the occurrence of different outcomes for a random variable. In the realm of continuous random variables, these distributions are defined via probability density functions (PDFs). The PDF provides a way to visualize and calculate where the random variable is likely to fall within a particular range.

Here are important aspects to understand about probability distributions for continuous variables:
  • The area under the curve of a PDF for an interval corresponds to the probability that the random variable falls within that interval.
  • A common example of a continuous probability distribution is the normal distribution, characterized by its bell-shaped curve.
  • Probability distributions must satisfy the property that the total area under the curve (over the entire space where the random variable can exist) equals one, ensuring that the probability of all possible outcomes sums up to one.
Integral of PDF
The integral of a probability density function (PDF) is a fundamental concept when working with continuous random variables. While the PDF gives us a way to model probabilities, it is the integral of the PDF over a specific interval that actually gives us the probability that the random variable falls within that range. This stems from the continuous nature of these variables and the infinite possibilities within any given range.

Here is how the integral of a PDF works:
  • If we have a PDF, say \( f(x) \), for a continuous random variable \( X \), the integral \( \int_{a}^{b} f(x) \, dx \) represents the probability that \( X \) falls between \( a \) and \( b \).
  • The integral of a PDF over the entire potential range of the variable must equal one, symbolizing that there is certainty (100% probability) that the variable will exist somewhere in that range.
  • The integral also highlights that the probability of observing any exact value (single point) of \( X \) is zero, reinforcing the concept that probabilities apply to intervals for continuous variables.

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Most popular questions from this chapter

The resistances of 50 resistors are measured and the results recorded as follows: $$ \begin{array}{cc} \hline \text { Resistance }(\Omega) & \text { Frequency } \\ \hline 5.0 & 17 \\ 5.5 & 12 \\ 6.0 & 10 \\ 6.5 & 6 \\ 7.0 & 5 \\ \hline \end{array} $$ Calculate the standard deviation of the measurements.

The data set \(A=\left\\{x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right\\}\) has a mean of \(\bar{x}\) and a standard deviation of \(\sigma .\) The data set \(B\) is \(\left\\{k x_{1}, k x_{2}, k x_{3}, \ldots, k x_{n}\right\\}\), the data set \(C\) is \(\left\\{x_{1}+k, x_{2}+k, x_{3}+k, \ldots, x_{n}+k\right\\}\) where \(k\) is a constant. (a) State the mean of set \(B\). (b) State the mean of set \(C\). (c) State the standard deviation of set \(B\). (d) State the standard deviation of set \(C\).

The probability of passing a module on the first attempt is \(0.9\). A student takes six modules. Calculate the probability that the student (a) passes five modules (b) passes all modules (c) is required to take two or more resits.

The temperature, \(T{ }^{\circ} \mathrm{C}\), of a freezer follows a normal distribution with mean \(-6{ }^{\circ} \mathrm{C}\) and standard deviation of \(2{ }^{\circ} \mathrm{C} .\) Calculate the probability that (a) \(T>-5\) (b) \(T<-7\) (c) \(-6

The probability that a machine has a lifespan of more than 7 years is \(0.85\). Twelve machines are chosen at random. Calculate the probability that (a) 10 have a lifespan of more than 7 years (b) 11 have a lifespan of more than 7 years (c) 10 or more have a lifespan of more than 7 years.
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