Question

Suppose that \(X\) is a continuous random variable. Explain why \(P(X=x)=0\). Which of the following probabilities are the same? Explain. $$ \begin{array}{ll} P(a

Short answer

For a continuous random variable, all interval probabilities are the same due to zero probability at specific points.

Step-by-step solution

Understanding Continuous Random Variables

A continuous random variable can take any value within a given range. For example, the random variable might represent temperatures, weights, or heights. This means the set of possible values is uncountably infinite.

Probability of a Single Point

In the case of a continuous random variable, the probability of the variable taking any single, specific value is zero. Mathematically, for a continuous random variable \( X \), \( P(X = x) = 0 \) because a single point has no length or area in the continuous sample space.

Interval Probabilities for Continuous Variables

For continuous random variables, probabilities are determined over intervals. This is because probabilities in continuous distributions are calculated as areas under the probability density function curve.

Comparing Interval Probabilities

Let's compare the given probabilities: \( P(a < X < b) \), \( P(a \leq X \leq b) \), \( P(a < X \leq b) \), and \( P(a \leq X < b) \). For continuous random variables, the probability at any specific point is zero, so the inclusion or exclusion of endpoints in an interval makes no difference to the probability.

Conclusion of Interval Equivalency

Therefore, for a continuous random variable, all given probabilities are the same: \( P(a < X < b) = P(a \leq X \leq b) = P(a < X \leq b) = P(a \leq X < b) \). This is because the probability of hitting the exact endpoints \( a \) or \( b \) is zero in each case.

Key concepts

Probability of a Single Point

When dealing with continuous random variables, an interesting outcome appears: the probability of a single, specific value occurring is actually zero. This might sound a bit puzzling at first. However, it makes sense when you consider that continuous random variables can take infinitely many possible values, just like any real number within a range.
To picture this, think about choosing a random number between 0 and 1. The odds of picking exactly 0.5 are practically impossible when there are infinitely many numbers to select from. Therefore, for any specific value, such as \( x \), the mathematical expression is \( P(X = x) = 0 \). It implies that any individual point has no probability weight in the continuum of possibilities, represented by a line, which has no length attached to a single point.

Interval Probabilities

Instead of single-point probabilities, continuous random variables focus on intervals. Interval probabilities are essential because they account for the occurrence of the variable within a range of values between two boundaries. This is because a range, unlike a single point, encompasses an area on the number line.
In mathematical terms, for any two numbers, \( a \) and \( b \), within the domain of a continuous random variable \( X \), the probability \( P(a < X < b) \) reflects the chance that \( X \) takes on a value between \( a \) and \( b \). Due to the properties of continuous variables, the probability is represented as an area under the curve of the Probability Density Function (PDF). This area provides a non-zero probability value indicative of the likelihood of \( X \) falling within that interval.

Probability Density Function

The Probability Density Function (PDF) is a crucial aspect of understanding continuous random variables. It describes how the probabilities are distributed over the values in the range of the random variable. Unlike a probability mass function (used for discrete variables), a PDF does not give probabilities directly, but rather densities.
For a continuous random variable \( X \), the PDF is denoted as \( f(x) \), determining the relative likelihood of \( X \) close to \( x \). The actual probability of any interval \( [a, b] \) is obtained by integrating the PDF over that interval:\[P(a < X < b) = \int_{a}^{b} f(x) \, dx\]The entire area under the PDF curve (from negative infinity to positive infinity) integrates to 1, symbolizing the total probability space. This underscores that probabilities for continuous variables are about areas, not points, aligning beautifully with the nature of continuous distributions.

Uncountably Infinite

The term "uncountably infinite" plays a significant role in understanding continuous random variables. In contrast to countably infinite sets, like the set of integers, uncountably infinite sets cannot be matched one-to-one with the natural numbers.
In the realm of continuous random variables, the values they can adopt form an uncountably infinite set. Take the real numbers between 0 and 1, for example; there are infinitely many and can't be listed in a sequence. This uncountable infinity underlines why the probability for a single point is zero and why area or interval probabilities are so crucial for continuous variables.
- There are more real numbers than integers, showcasing the vastness of uncountable sets. - This concept helps explain the contrasts between discrete and continuous distributions, particularly how probabilities are approached differently.