Continuous Random Variable
A continuous random variable is a variable that can take on an uncountably infinite number of values within a given range, and its outcomes are defined in a sample space that is not countable. An example of a continuous random variable is the time it takes to perform a task, such as reading an article, which can theoretically be any positive real number. Unlike discrete random variables, which have a finite or countable number of possible outcomes, continuous random variables require a different approach to probability. Specifically, the probability of a continuous random variable taking on any specific value is zero; therefore, probabilities are calculated over intervals using the concept of probability density functions (PDFs).
To determine the likelihood of a continuous random variable falling within a particular range, one integrates the PDF over that range. This leads to expressions like \( \text{Prob}(a < X < b) \), which gives the probability that the variable \( X \) lies between two values \( a \) and \( b \). A common model for such distributions is the negative exponential distribution, often used to represent time-related events. Understanding continuous random variables is crucial for various fields, including statistics, engineering, and economics, as they allow for modeling and analysis of real-world phenomena where outcomes are not easily categorized.
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Continuous Random Variable
A continuous random variable is one defined on an uncountably infinite sample space. A sample space is a set of possible outcomes of some procedure or trial where there is more than one possible outcome. An example would be the length of time it takes to read this article. The time could be measured to the nearest minute to yield a discrete random variable. Allowing the possibility of infinite reading time, there are many possible outcomes of any number greater than zero.
Some notion of probability must be associated with the sample space. A random variable is a function that maps the sample space onto a number. In this case, time elapsed is associated with a number. As we are mapping an event with an associated probability we induce some notion of probability onto the numbers we obtain. The probability that a random variable X takes on a specific value is expressed as x Prob(X = x).
The key distinction between continuous and discrete random variables is that for continuous random variables the sample space is infinite and uncountable. The random variable can be turned into a discrete random variable by recording the time in rounded units, such as minutes or microseconds. However, even if the time available were restricted to a maximum of one hour, there are still uncountably many real numbers between 0 and 60.
The key implication of dealing with an uncountable sample space is that the probability associated with any specifically realised number is zero. A rather approximate analogy is the event of rolling a fair die, which has a finite countable sample space; you obtain dot, 2 dots,…,6 dots. Associated with this is the random variable X, which denotes the number of dots (1,2,…6) and has associated the probability 1/6 that any particular number is seen on any given die throw. If the number of sides of die is increased the probability of any particular event decreases with the number of sides. For example, with 12 sides the probability is 1/12. In rolling a die with an uncountably infinite number of sides, the probability of the die landing on any one side is zero.
Rather than Prob(X = x), Prob(a<X<b) is used to determine the probability that the random variable X lies in a subset of the sample space between real numbers a and b. This subset E can be defined formally as E: {a < x < b}. It is possible to find the probability that event E occurs. The experiment is conducted and the random variable X takes a number between a and b by using
Here f(x) denotes any function that acts as a probability density function. This function can model the notion of probability associated with the random variable X. If we integrate this function over the range of interest we obtain the probability that a random variable X lies in that range.
One common choice of probability model for the time it takes to read this article would be the so-called negative exponential distribution. For some parameter
and random variable
the probability density function is given by:
If, for example, the parameter
we could find
=
Bibliography
Blitzstein, J. K., and J. Hwant. Introduction to Probability. Boca Raton, FL: CRC, 2015.
Mendenhall, William, Robert J. Beaver, Barbara M. Beaver. Introduction to Probability and Statistics. Boston: Cengage, 2013.
Pishro-Nik, Hossein. Introduction to Probability, Statistics, and Random Processes. Kappa Research, 2014.