As a portfolio manager I evaluate the stock-weights in a portfolio and I want only those stocks that can perform better during inflation. By probability distribution, I can distribute their probability of performing during inflation by reviewing the historical data. Hence, we can easily collate the probable returns. Probability distributions is covered under Quants-I for the FRM-I curriculum and the total quantitative analysis has 20% weightage in the FRM-I exam.

## Probability Distribution

A business analyst can access data, analyze information and process them to meaningful insights. So it is understood that any analytics depends heavily on the appropriate usage of probability distribution to accurately represent the uncertainty, randomness and variability of the problem. An analyst’s inappropriate use of probability distributions can lead to failure of risk analysis models. It majorly happens either due to inadequate understanding of theory behind probability or failing to understand the vast damages it can create due to inaccurate usage of distributions. For such an elaborate process to be analyzed understanding the concept becomes essential.

Practically probability distributions and scenario analysis is done in a business to predict future levels of sales. It is impossible to predict the exact value of future sales level; businesses however still need to plan for future events. Using probability distribution can help a company frame its possible future values in terms of possible sales level and predict best and worst case scenario. Thus, a company can make its business plans on the likely scenario but still be aware of the alternative possibilities, as well. For better understanding of probability distributions, it’s important to understand variables and random variables. Let’s start with understanding what a variable is.

A variable is a symbol (A, B, x, y, etc.) that can take on any of a specified set of values and Random Variables are a function or a rule which maps each event in a sample space to real numbers. For example: If you are tossing a coin: you can get Heads or Tails. Let's consider values of Headsas0 and Tailsas1 and we have a Random Variable "X" For Random Variable X the possible values would be 0 and 1, in other words X = {0, 1}

Random Variable also has two types. The first one being: Discrete Random Variables which is the set of all possible values of the outcome. For Example: If an analyst is asked about the credit cards owned by an individual, it can be counted anything as {0, 1, 2, 3, 4, 5, 6, 7, 8, and 9}. The second one being Continuous Random Variables: The set of possible values taken by a continuous random variable falls in an interval. For Example: The Salary received by the set of people working in the same profile.

We have talked about both probability distribution and variables; now let’s take an example, to understand the relationship between random variables and probability distributions.

Example: Suppose a coin is flipped two times. This statistical experiment can have four possible results: HH, HT, TH, and TT. Now, we consider the variable X which represents the number of Heads that may come from this experiment. The variable X can have the following values 0, 1 or 2. In this example, X is a random variable; because its value is determined by the outcome of a statistical experiment.

Now, considering the same coin flip experiment described above, the table below associates each outcome with its probability, is an example of a probability distribution.

##### Probability
0 0.25
1 0.50
2 0.25

Both Discrete Probability Distribution and Continuous Probability Distribution have further sub division; let’s have a quick look at Binomial Distribution and Normal Distribution:

The Binomial Distribution is widely used to test statistical probabilities and significance. It’s a good way of visually detecting unexpected values and a useful tool in determining permutations, combinations, and probabilities, where the outcomes can be broken down into two probabilities. Common uses of binomial distributions in business include quality control, public opinion surveys, medical research, and insurance problems. It can be applied to complex processes such as sampling items in factory production lines or to estimate percentage failure rates of products and components.

In statistics, an average distribution of values states that, when plotted on a graph, resembles the shape of a bell. The curve formed by a Normal Distribution is called a normal curve, bell curve, normal distribution curve or normal probability curve. It is useful because it’s easy to work with mathematically. The amount by which values vary from one another defines the shape of the normal curve. If most of the values are similar, then the normal curve is a tall and thin bell shape. If there is a lot of variation among the values, then the normal curve is a short and wide bell shape. One measure of the variation among values is called the standard deviation.

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