statistics

Question

Tell if each variable is continuous or discrete.

a. Tonnage carried by an oil tanker at sea.
b. Wind velocity at 7 o’clock this morning.
c. Number of text messages you received yesterday.

Answer 1

a and b are continuous since within a reasonable interval (e.g. (0,20) for a), any real number is possible.

 

c is discrete as there are finite gaps between consecutive possible values, which in this case must be non-negative integers 0,1,2,3,...

Answer 2

A continuous random variable is one that can can take on a range of values. That is, such a random variable can take any real number (possibly in a certain range). That means that the tonnage is a continuous random variable: the oil tanker can carry 10 tons, 11, 10.5, 10.2 tons, and so forth. The same is true about wind velocity (although it would be better described with two continuous random variables rather than one: speed and azimuth). Both the speed and the azimuth (a way of specifying direction with a number) are continuous: the speed can be 10 miles per hour, 10.1, 10.01, and so forth, and the azimuth can be 45 degrees, 45.1 degrees, and so forth. For a continuous random variable, if it can take on a value, then it should be able to take on values very close to it.

 

A discrete random variable takes on values that are separated from each other. That is, such a random variable may be allowed to take on the values 1 and 2, but nothing in between. A variable taking on only integer values would be discrete. The key to recognizing a discrete random variable is that for a given value it can take, there is a "next" and a "previous" value for it. For instance, the amount of money in a cash register is a discrete random variable. If there is more than $45.41 in it, then the next possible value is $45.42; $45.4101 is not a possible value because currency is only divided into pennies and not further. The number of rainy days in a given town in 2015 is a clearer example: either there is one rainy day, or 2, or 3, but not 1.5, at least if a rainy day is defined as one on which any rain falls (or in another clear way).

 

See if you can figure out part c from this distinction.

 

Why is the distinction important? A discrete random variable can be equal to a certain exact value with a positive probability. For instance, there is a positive probability that there will be exactly 109 rainy days in Charlotte in 2015. However, the probability that there will fall exactly 41.509 inches of rain in 2015 is zero - you can see why that is: if this really happened with a positive probability, then 41.510 inches will need to be assigned a positive probability, as will 41.50901, as will any other real number in this range. There are too many real numbers to assign a positive probability to each one: if you do, you will find that the total probability exceeds 1, no matter how small the probability you assign. However, the probability that there will be between 41.5 and 41.6 inches is positive. The same is true for the range 41.50 to 41.51, and for 41.509 to 41.510, and so forth. As the interval narrows, the probability decreases. If each value in a narrow region is roughly equally likely, then the probability of finding the random variable is proportional to the error you are willing to tolerate when deciding whether to call your value close enough to the one you are looking for. The coefficient of proportionality is an important number; it tells you how likely the random variable is to be in one range, relative to another. There is actually just one coefficient of proportionality for each value of the random variable (it would be a limit of these coefficients as the range gets narrower, if you know calculus), and if you plot this coefficient versus the value of the random variable, you get the probability density function.

 

Hopefully, you learned a bit more than just the answer to your question. Good luck in your Statistics course.