{% hint style="success" %} A random variable is a variable that is subject to randomness, which means it can take on different values. {% endhint %}
- In statistics, it is normal to use an
Xto denote a random variable. - The random variable takes on different values depending on the situation.
- Each value of the random variable has a probability or percentage associated with it.
A discrete random variable is a variable that represents numbers found by counting. For example, number of marbles in a jar, number of students present or number of heads when tossing two coins.
{% hint style="info" %} X is discrete because the numbers that X represents are isolated points on the number line. {% endhint %}
The number of heads that can come up when tossing two coins is a discrete random variable because heads can only come up a certain number of times: 0, 1, or 2. Also, we want to know the probability associated with each value of the random values.
A probability distribution has all the possible values of the random variable and the associated probabilities.
{% hint style="success" %} When we have to use intervals for our random variable or all values in an interval are possible, we call it a continuous random variable. {% endhint %}
Thus, continuous random variables are random variables that are found from measuring - like the height of a group of people or distance traveled while grocery shopping or student test scores.
{% hint style="info" %} In this case, X is continuous because X represents an infinite number of values on the number line. {% endhint %}
Like the coin example, the random variable (in this case, the intervals) would have certain probabilities or percentages associated with it. And this would be a probability distribution for the test scores.
In the study of probability, we are interested in finding the probabilities associated with each value of these random variables.
Sum of Probabilities for a Distribution
On the above, in each table, the sum of all probabilities add up to 1 or 100%. However, for continuous random variables, we can construct a histogram of the table with relative frequencies, and the area under the histogram is also equal to 1.