A random variable takes the value of a number that is drawn from a known distribution. Once we draw the variable, the number we have drawn is said to be a realization of the random variable.

If we draw a number from a distribution infinitely many times, the average of those draws (realizations) will be the Mean of the distribution. Variance is the expected value of squared deviation from the mean. Informally, it measures how far a set of random draws from a distribution are spread out from their mean.

Normal Distribution is a pattern for the probability distribution of a set of data which follows a symmetric bell-shaped curve. A normal distribution is characterized by its mean \(\boldsymbol\mu\) and variance \(\boldsymbol{\sigma^2}\). The probability of different values occurring is symmetric around \(\boldsymbol\mu\). Note that the smaller the variance \(\boldsymbol{\sigma^2}\) is, the more centered the bell-shaped curve looks. Moreover, the drawn number will lie within \(\boldsymbol{\mu - 1.96\sqrt{\sigma^2}}\) and \(\boldsymbol{\mu + 1.96\sqrt{\sigma^2}}\) with probability of 95%.

Updating Belief Using New Information: Suppose you know that a random variable is drawn from a normal distribution. This is called the prior distribution of the random variable. Suppose you do not know the realized value. However, you receive forecasts about the realized value that are drawn from a normal distribution with mean equaling the realized value and some known variance. Then, you can update your belief about how the realized value is distributed. This distribution should be normally distributed and its mean will be a weighted average of the prior mean and the observed forecasts, where the weights will depend on the variances of the prior distribution and the distributions of forecasts. This mean will be your prediction for the realized value.