Statistical Inference

In real life, we work with data that are affected by randomness, and we need to extract information and draw conclusions from the data. The randomness might come from a variety of sources. Here are two examples of such situations:

1.Suppose that we would like to predict the outcome of an election. Since we cannot poll the entire population, we will choose a random sample from the population and ask them who they plan to vote for. In this experiment, the randomness comes from the sampling. Note also that if our poll is conducted one month before the election, another source of randomness is that people might change their opinions during the one month period.

2. In a wireless communication system, a message is transmitted from a transmitter to a receiver. However, the receiver receives a corrupted version (a noisy version) of the transmitted signal. The receiver needs to extract the original message from the received noisy version. Here, the randomness comes from the noise.

Examples like these are abundant. Dealing with such situations is the subject of the field of statistical inference.

Statistical inference is a collection of methods that deal with drawing conclusions from data that are prone to random variation. 

Clearly, we use our knowledge of probability theory when we work on statistical inference problems. However, the big addition here is that we need to work with real data. The probability problems that we have seen in this book so far were clearly defined and the probability models were given to us. For example, you might have seen a problem like this: 

Random Sampling