Quantitative Methods  Common Probability Distribution Calculations
Cumulative Distribution Functions
A cumulative distribution function or CDF, expresses a probability's function in terms of lowest to highest value, by giving the probability that a random variable X is less than or equal to a particular value x. Expressed in shorthand, the cumulative distribution function is P(X < x). A cumulative distribution function is constructed by summing up, or cumulating all values in the probability function that are less than or equal to x. The concept is similar to the cumulative relative frequency covered earlier in this study guide, which computed values below a certain point in a frequency distribution.
Example: Cumulative Distribution Function
For example, the following probability distribution includes the cumulative function.
X = x  P(X = x)  P(X < x) or cdf 
< 12  0.15  0.15 
12 to 3  0.15  0.30 
3 to 4  0.25  0.55 
4 to 10  0.25  0.80 
> 10  0.2  1.0 
From the table, we find that the probability that x is less than or equal to 4 is 0.55, the summed probabilities of the first three P(X) terms, or the number found in the cdf column for the third row, where x < 4. Sometimes a question might ask for the probability of x being greater than 4, for which this problem is 1  P(x < 4) = 1  0.55 = 0.45. This is a question most people should get  but one that will still have too many people answering 0.55 because they weren't paying attention to the "greater than".
Discrete Uniform Random Variable
A discrete uniform random variable is one that fulfills the definition of "discrete", where there are a finite and countable number of terms, along with the definition of "uniform", where there is an equally likely probability that the random variable X will take any of its possible values x. If there are n possible values for a discrete uniform random variable, the probability of a specific outcome is 1/n.
Example: Discrete Uniform Random Variable
Earlier we provided an example of a discrete uniform random variable: a random day is oneseventh likely to fall on a Sunday. To illustrate some examples on how probabilities are calculated, take the following discrete uniform distribution with n = 5.
X = x  P(X = x)  P(X < x) 
2  0.2  0.2 
4  0.2  0.4 
6  0.2  0.6 
8  0.2  0.8 
10  0.2  1.0 
According to the distribution above, we have the probability of x = 8 as 0.2. The probability of x = 2 is the same, 0.2.
Suppose that the question called for P(4 < X < 8). The answer would be the sum of P(4) + P(6) + P(8) = 0.2 + 0.2 + 0.2 = 0.6.
Suppose the question called for P(4 < X < 8). In this case, the answer would omit P(4) and P(8) since it's less than, NOT less than or equal to, and the correct answer would be P(6) = 0.2. The CFA exam writers love to test whether you are paying attention to details and will try to trick you  the probability of such tactics is pretty much a 1.0!
Binomial Random Variable
Binomial probability distributions are used when the context calls for assessing two outcomes, such as "success/failure", or "price moved up/price moved down". In such situations where the possible outcomes are binary, we can develop an estimate of a binomial random variable by holding a number of repeating trials (also known as "Bernoulli trials"). In a Bernoulli trial, p is the probability of success, (1  p) is the probability of failure. Suppose that a number of Bernoulli trails are held, with the number denoted by n. A binomial random variable X is defined as the numberof successes in n Bernoulli trials, given two simplifying assumptions: (1) the probabilityp of success is the same for all trials and (2) the trials are independent of each other.
Thus, a binomial random variable is described by two parameters: p (the probability of success of one trial) and n (the number of trials). A binomial probability distribution with p = 0.50 (equal chance of success or failure) and n = 4 would appear as:
x (# of successes)  p(x)  cdf, P(X < x) 
0  0.0625  0.0625 
1  0.25  0.3125 
2  0.375  0.6875 
3  0.25  0.9325 
4  0.0625  1.0000 
The reference text demonstrates how to construct a binomial probability distribution by using the formula p(x) = (n!/(n  x)!x!)*(p^{x})*(1  p)^{nx}. We used this formula to assemble the above data, though the exam would probably not expect you to create each p(x); it would probably provide you with the table, and ask for an interpretation. For this table, the probability of exactly one success is 0.25; the probability of three or fewer successes is 0.9325 (the cdf value in the row where x = 3); the probability of at least one is 0.9325 (1  P(0)) = (1  0.0625) = 0.9325.
Calculations
The expected value of a binomial random variable is given by the formula n*p. In the example above, with n = 4 and p = 0.5, the expected value would be 4*0.5, or 2.
The variance of a binomial random variable is calculated by the formula n*p*(1  p). Using the same example, we have variance of 4*0.5*0.5 = 1.
If our binomial random variable still had n = 4 but with a greater predictability in the trial, say p = 9, our variance would reduce to 4*0.9*0.1 = 0.36. For successive trials (i.e. for higher n), both mean and variance increase but variance increases at a lower rate  thus the higher the n, the better the model works at predicting probability.
Creating a Binomial Tree
The binomial tree is essentially a diagram showing that the future value of a stock is the product of a series of up or down movements leading to a growing number of possible outcomes. Each possible value is called a node.
Figure 2.9: Binomial Tree 
Continuous Uniform Distribution
A continuous uniform distribution describes a range of outcomes, usually bound with an upper and lower limit, where any point in the range is a possibility. Since it is a range, there are infinite possibilities within the range. In addition, all outcomes are all equally likely (i.e. they are spread uniformly throughout the range).
To calculate probabilities, find the area under a pdf curve such as the one graphed here. In this example, what is the probability that the random variable will be between 1 and 3? The area would be a rectangle with a width of 2 (the distance between 1 and 3), and height of 0.2, 2*0.2 = 0.4.
What is the probability that x is less than 3? The rectangle would have a width of 3 and the same height: 0.2. 3*0.2 = 0.6

Personal Finance
How To Choose A Financial Advisor
Many advisors display similar skillsets that can make distinguishing between them difficult. The following guidelines can help you better understand their qualifications and services. 
Investing
Asset Manager Ethics: Investment Process and Actions
Managers, in developing their investment process, need to determine some “general rules” that make it meaningful. We offer six. 
Professionals
Career Advice: Financial Analyst Vs. Investment Banker
Read an indepth comparison about working as a Financial Analyst vs. working as an Investment Banker, two highly prestigious business careers. 
Professionals
Advisors: Which Certifications Are Essential?
The right advisor credentials can make all the difference, but wading through some 100 certifications can be a challenge. Here's some help. 
Investing Basics
Asset Manager Ethics: Valuation Is A Tricky Business
Asset managers must accurately represent all of a clients assets in the client portfolio. This can be tricky for unique and hardtovalue assets. 
Personal Finance
Top 10 Most Valuable Sports Teams in 2015
Cleats, pads and profits: we take a look at the top 10 most valuable sports teams in the world. 
Professionals
Chinese Slowdown Affects Iron Ore Market
The Chinese economy's ongoing slowdown is having a major impact on iron ore demand. 
Personal Finance
Invest in Costco? First Understand Its Balance Sheet
A strong balance sheet sets a company apart and boosts investor confidence. How healthy is Costco based on an analysis of its balance sheets from the last two years? 
Investing Basics
Brokers and RIAs: One and the Same?
Brokers and registered investment advisors have some key differences. Here's what you need to know. 
Professionals
DCF Vs. Comparables: Which One To Use
DCF and Comparables models are widely used in equity valuation. We explain the pros and cons of each method.

Personal Financial Advisor
Professionals who help individuals manage their finances by providing ... 
CFA Institute
Formerly known as the Association for Investment Management and ... 
Chartered Financial Analyst  CFA
A professional designation given by the CFA Institute (formerly ... 
Security Analyst
A financial professional who studies various industries and companies, ...

What are the differences between a Chartered Financial Analyst (CFA) and a Certified ...
The differences between a Chartered Financial Analyst (CFA) and a Certified Financial Planner (CFP) are many, but comes down ... Read Full Answer >> 
How do I become a Chartered Financial Analyst (CFA)?
According to the CFA Institute, a person who holds a CFA charter is not a chartered financial analyst. The CFA Institute ... Read Full Answer >> 
What types of positions might a Chartered Financial Analyst (CFA) hold?
The types of positions that a Chartered Financial Analyst (CFA) is likely to hold include any position that deals with large ... Read Full Answer >> 
Who benefits the most from prepaid expenses?
Prepaid expenses benefit both businesses and individuals. Prepaid expenses are the types of expenses that are bought or paid ... Read Full Answer >> 
If I am looking to get an Investment Banking job. What education do employers prefer? ...
If you are looking specifically for an investment banking position, an MBA may be marginally preferable over the CFA. The ... Read Full Answer >> 
Can I still pass the CFA Level I if I do poorly in the ethics section?
You may still pass the Chartered Financial Analysis (CFA) Level I even if you fare poorly in the ethics section, but don't ... Read Full Answer >>