- Examples
- Continuous variables and discrete variables
- Continuous variables exercise
- Solution
- Exercise of
- -Exercise 1 of probabilities
- Solution
- -Exercise 2 of probabilities
- References
The continuous variable is one that can take an infinite number of numerical values between two given values, even if those two values are arbitrarily close. They are used to describe measurable attributes; for example height and weight. The values that a continuous variable takes can be rational numbers, real numbers or complex numbers, although the latter case is less frequent in statistics.
The main characteristic of continuous variables is that between two rational or real values another can always be found, and between that other and the first another value can be found, and so on indefinitely.
Figure 1. The curve represents a continuous distribution and the bars a discrete one. Source: pixabay
For example, suppose the variable weight in a group where the heaviest weighs 95 kg and the lowest weighs 48 kg; that would be the range of the variable and the number of possible values is infinite.
For example between 50.00 kg and 50.10 kg can be 50.01. But between 50.00 and 50.01 can be the measure 50.005. That is a continuous variable. On the other hand, if a precision of a single decimal were established in the possible measurements of weight, then the variable used would be discrete.
Continuous variables belong to the category of quantitative variables, because they have a numerical value associated with them. With this numerical value it is possible to carry out mathematical operations ranging from arithmetic to infinitesimal calculation methods.
Examples
Most of the variables in physics are continuous variables, among them we can name: length, time, speed, acceleration, energy, temperature and others.
Continuous variables and discrete variables
In statistics, various types of variables can be defined, both qualitative and quantitative. Continuous variables belong to the latter category. With them it is possible to carry out arithmetic and calculation operations.
For example, the variable h, corresponding to people with height between 1.50 m and 1.95 m, is a continuous variable.
Let's compare this variable with this one: the number of times a coin toss comes up heads, which we will call n.
The variable n can take values between 0 and infinity, however n is not a continuous variable since it cannot take the value 1.3 or 1.5, because between values 1 and 2 there is no other. This is an example of a discrete variable.
Continuous variables exercise
Consider the following example: a machine produces matchsticks and packs them in its box. Two statistical variables are defined:
The nominal match length is 5.0 cm with a tolerance of 0.1 cm. The number of matches per box is 50 with a tolerance of 3.
a) Indicate the range of values that L and N can take.
b) How many values can L take?
c) How many values can n take?
State in each case whether it is a discrete or continuous variable.
Solution
The values of L are in the range; that is, the value of L is in the interval and the variable L can take infinite values between these two measurements. It is then a continuous variable.
The value of variable n is in the interval. The variable n can only take 6 possible values in the tolerance interval, it is then a discrete variable.
Exercise of
If, in addition to being continuous, the values taken by the variable have a certain probability of occurrence associated with them, then it is a continuous random variable. It is very important to distinguish if the variable is discrete or continuous, since the probabilistic models applicable to one and the other are different.
A continuous random variable is completely defined when the values that it can assume, and the probability that each one of them has of happening, are known.
-Exercise 1 of probabilities
The matchmaker makes them in such a way that the length of the sticks is always between the values 4.9 cm and 5.1 cm, and zero outside these values. There is a probability of obtaining a stick that measures between 5.00 and 5.05 cm, although we could also extract one of 5,0003 cm. Are these values equally likely?
Solution
Suppose the probability density is uniform. The probabilities of finding a match with a certain length are listed below:
-That a match is in the range has probability = 1 (or 100%), since the machine does not draw matches outside of those values.
-Finding a match that is between 4.9 and 5.0 has probability = ½ = 0.5 (50%), since it is half the range of lengths.
-And the probability that the match has length between 5.0 and 5.1 is also 0.5 (50%)
-It is known that there are no match sticks that have a length between 5.0 and 5.2. Probability: zero (0%).
Probability of finding a toothpick in a certain range
Now let us observe the following probabilities P of obtaining sticks whose length is between l 1 and l 2:
-P that a match has a length between 5.00 and 5.05 is denoted as P ():
-P that the hill has length between 5.00 and 5.01 is:
-P that the hill has a length between 5,000 and 5,001 is even less:
If we keep decreasing the interval to get closer and closer to 5.00, the probability that a toothpick is exactly 5.00 cm is zero (0%). What we do have is the probability of finding a match within a certain range.
Probability of finding multiple toothpicks in a given range
If the events are independent, the probability that two toothpicks are in a certain range is the product of their probabilities.
-The probability that two chopsticks are between 5.0 and 5.1 is 0.5 * 0.5 = 0.25 (0.25%)
-The probability that 50 toothpicks are between 5.0 and 5.1 is (0.5) ^ 50 = 9 × 10 ^ -16, that is to say almost zero.
-The probability that 50 toothpicks are between 4.9 and 5.1 is (1) ^ 50 = 1 (100%)
-Exercise 2 of probabilities
In the previous example, the assumption was made that the probability is uniform in the given interval, however this is not always the case.
In the case of the actual machine that produces the toothpicks, the chance that the toothpick is at the center value is greater than it is at one of the extreme values. From a mathematical point of view this is modeled with a function f (x) known as the probability density.
The probability that the measure L is between a and b is calculated using the definite integral of the function f (x) between a and b.
As an example, suppose that we want to find the function f (x), which represents a uniform distribution between the values 4.9 and 5.1 from exercise 1.
If the probability distribution is uniform, then f (x) is equal to the constant c, which is determined by taking the integral between 4.9 and 5.1 of c. Since this integral is the probability, then the result must be 1.
Figure 2. Uniform probability density. (Own elaboration)
Which means that c is worth 1 / 0.2 = 5. That is, the uniform probability density function is f (x) = {5 if 4.9≤x≤5.1 and 0 outside this range. A uniform probability density function is shown in Figure 2.
Note how in intervals of the same width (for example 0.02) the probability is the same at the center as at the end of the range of the continuous variable L (toothpick length).
A more realistic model would be a probability density function like the following:
Figure 3. Non-uniform probability density function. (Own elaboration)
In figure 3 it can be seen how the probability of finding toothpicks between 4.99 and 5.01 (width 0.02) is greater than that of finding toothpicks between 4.90 and 4.92 (width 0.02)
References
- Dinov, Ivo. Discrete Random Variables and Probability Distributions. Retrieved from: stat.ucla.edu
- Discrete and Continuous Random Variables. Retrieved from: ocw.mit.edu
- Discrete Random Variables and Probability Distributions. Retrieved from: homepage.divms.uiowa.edu
- H. Pishro. Introduction to Probability. Recovered from: probability course.com
- Mendenhall, W. 1978. Statistics for Management and Economics. Grupo Editorial Iberoamericana. 103-106.
- Random Variables Problems and Probability Models. Recovered from: ugr.es.
- Wikipedia. Continuous variable. Recovered from wikipedia.com
- Wikipedia. Statistics variable. Recovered from wikipedia.com.