Business Analytics > Probability, Risk & Sampling > Random Variable and its Probability
Random Variable
A random variable is a numerical description of the outcome of an experiment. Formally, a random variable assigns a real number to each element of a sample space. If we have categorical outcomes, we can associate an arbitrary numerical value to them.
For example, if a consumer likes a product in a market, we might assign this outcome a value of 1; if the consumer dislikes the product, we might assign this outcome a value of 0.
Random variables are usually denoted by capital italic letters, such as X or Y. Random variables may be discrete or continuous. A discrete random variable is one for which the number of possible outcomes can be counted and it is full number. A continuous random variable has outcomes over one or more continuous intervals of real numbers.
Probability Distribution
A probability distribution is the probability of each possible value of a random variable may have.
Suppose a shop records the number of customers who enter the store in one hour. The random variable X represents the number of customers. Note that, he observed it for several days and calculated proportion which can be regarded as experimental probability:
|
X (Customers) |
0 |
1 |
2 |
3 |
|
P(X) |
0.1 |
0.3 |
0.4 |
0.2 |
Here, each value of X has a probability, and the total probability adds up to 1. This table is a probability distribution and helps the business estimate customer flow and plan staffing.
Suppose a company studies the time (in minutes) a customer waits to be served at a bank counter. Let X represent the waiting time as follows:
X is a continuous random variable because it can take any value within a range i.e., 2.5 minutes, 3.2 minutes, 4.75 minutes.
Assume the waiting time follows a normal distribution with:
- Mean (average) waiting time = 5 minutes
- Standard deviation = 1 minute
The probability that a customer waits between 4 and 6 minutes is found using the normal distribution curve:
P(4<X<6)=?
This probability is represented by the area under the curve between 4 and 6 minutes.
In business, it helps the bank decide how many service counters are needed to reduce waiting time and improve customer satisfaction.
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