Knowledge in Maths

How to find the mean of the probability distribution: Steps

How to find the mean of the probability distribution: StepsStep 1: Convert all the percentages to decimal probabilities. For example: ...Step 2: Construct a probability distribution table. ...Step 3: Multiply the values in each column. ...Step 4: Add the results from step 3 together.


KEY TAKEAWAYS OF PROBABILITY DISTRIBUTION-A probability distribution depicts the expected outcomes of possible values for a given data generating process.Probability distributions come in many shapes with different characteristics, as defined by its mean, standard deviation, skewness, and kurtosis.Investors use probability distributions to anticipate returns on assets such as stocks over time and to hedge their risk.

Types of Probability Distributions

Types of Probability DistributionsThere are many different classifications of probability distributions. Some of them include the normal distribution, chi square istribution, binomial distribution and Poisson distribution. The different probability distributions serve different purposes and represent different data generation processes. The binomial distribution, for example, evaluates the probability of an event occurring several times over a given number of trials and given the event's probability in each trial. and may be generated by keeping track of how many free throws a basketball player makes in a game, where 1 = a basket and 0 = a miss. Another typical example would be to use a fair coin and figuring the probability of that coin coming up heads in 10 straight flips. A binomial distribution is discrete, as opposed to continuous, since only 1 or 0 is a valid response.The most commonly used distribution is the normal distribution, which is used frequently in finance, investing, science, and engineering. The normal distribution is fully characterized by its mean and standard deviation, meaning the distribution is not skewed and does exhibit kurtosis. This makes the distribution symmetric and it is depicted as a bell-shaped curve when plotted. A normal distribution is defined by a mean (average) of zero and a standard deviation of 1.0, with a skew of zero and kurtosis = 3. In a normal distribution, approximately 68 percent of the data collected will fall within +/- one standard deviation of the mean; approximately 95 percent within +/- two standard deviations; and 99.7 percent within three standard deviations. Unlike the binomial distribution, the normal distribution is continuous, meaning that all possible values are represented (as opposed to just 0 and 1 with nothing in between).Probability Distributions Used in InvestingStock returns are often assumed to be normally distributed but in reality, they exhibit kurtosis with large negative and positive returns seeming to occur more than would be predicted by a normal distribution. In fact, because stock prices are bounded by zero but offer a potential unlimited upside, the distribution of stock returns has been described as log normal. This shows up on a plot of stock returns with the tails of the distribution having greater thickness.Probability distributions are often used in risk management as well to evaluate the probability and amount of losses that an investment portfolio would incur based on a distribution of historical returns. One popular risk management metric used in investing is value-at-risk (VaR). VaR yields the minimum loss that can occur given a probability and time frame for a portfolio. Alternatively, an investor can get a probability of loss for an amount of loss and time frame using VaR. Misuse and overreliance on var has been implicated as one of the major causes of the 2008 financial crisis.Example of a Probability DistributionAs a simple example of a probability distribution, let us look at the number observed when rolling two standard six-sided dice. Each die has a 1/6 probability of rolling any single number, one through six, but the sum of two dice will form the probability distribution depicted in the image below. Seven is the most common outcome (1+6, 6+1, 5+2, 2+5, 3+4, 4+3). Two and twelve, on the other hand are far less likely (1+1 and 6+6).


What Are Statistics?Statistics is a form of mathematical analysis that uses quantified models, representations and synopses for a given set of experimental data or real-life studies. Statistics studies methodologies to gather, review, analyze and draw conclusions from data. Some statistical measures include the following:MeanSkewnessKurtosisVarianceAnalysis of variance

Understanding Statistics

Understanding StatisticsStatistics is a term used to summarize a process that an analyst uses to characterize a data set. If the data set depends on a sample of a larger population, then the analyst can develop interpretations about the population primarily based on the statistical outcomes from the sample. Statistical analysis involves the process of gathering and evaluating data and then summarizing the data into a mathematical form.Statistics is used in various disciplines such , business, physical and social sciences, humanities, government, and manufacturing. Statistical data is gathered using a sample procedure or other method. Two types of statistical methods are used in analyzing data: descriptive statistics and inferential statistics. Descriptive statistics are used to synopsize data from a sample exercising the mean or standard deviation. Inferential statistics are used when data is viewed as a subclass of a specific population.

KEY TAKEAWAYS of statistics

KEY TAKEAWAYSStatistics studies methodologies to gather, review, analyze, and draw conclusions from data.There are many different types of statistics pertaining to which situation you need to analyze.Statistics are used to make better-informed business decisions

Types of Statistics

Types of StatisticsStatistics is a general, broad term, so it's natural that under that umbrella there exist a number of different models.MeanA mean is the mathematical average of a group of two or more numerals. The mean for a specified set of numbers can be computed in multiple ways, including the arithmetic mean, which shows how well a specific commodity performs over time, and the geometric mean, which shows the performance results of an investor’s portfolio invested in that same commodity over the same period.Regression AnalysisRegression analysis determines the extent to which specific factors such as interest rates, the price of a product or service, or particular industries or sectors influence the price fluctuations of an asset. This is depicted in the form of a straight line called linear regression.SkewnessSkewness describes the degree a set of data varies from the standard distribution in a set of statistical data. Most data sets, including commodity returns and stock prices, have either positive skew, a curve skewed toward the left of the data average, or negative skew, a curve skewed toward the right of the data average.KurtosisKurtosis measures whether the data are light-tailed (less outlier-prone) or heavy-tailed (more outlier-prone) than the normal distribution. Data sets with high kurtosis have heavy tails, or outliers, which implies greater investment risk in the form of occasional wild returns. Data sets with low kurtosis have light tails, or lack of outliers, which implies lesser investment risk.VarianceVariance is a measurement of the span of numbers in a data set. The variance measures the distance each number in the set is from the mean. Variance can help determine the risk an investor might accept when buying an investment.Ronald Fisher developed the analysis of variance method. It is used to decide the effect solitary variables have on a variable that is dependent. It may be used to compare the performance of different stocks over time.


Variables are used to store information to be referenced and manipulated in a computer program. They also provide a way of labeling data with a descriptive name, so our programs can be understood more clearly by the reader and ourselves. It is helpful to think of variables as containers that hold information. Their sole purpose is to label and store data in memory. This data can then be used throughout your program.

Naming variables

Naming variables -is known as one of the most difficult tasks in computer programming. When you are naming variables, think hard about the names. Try your best to make sure that the name you assign your variable is accurately descriptive and understandable to another reader. Sometimes that other reader is yourself when you revisit a program that you wrote months or even years earlier.When you assign a variable, you use the =symbol. The name of the variable goes on the left and the value you want to store in the variable goes on the right.irb :001 > first_name = 'Joe' => "Joe" Here we've assigned the value 'Joe', which is a string, to the variable first_name. Now if we want to reference that variable, we can.irb :002 > first_name => "Joe" As you can see, we've now stored the string 'Joe' in memory for use throughout the program.Note: Make sure you don't confuse the assignment operator (=) with the equality operator (==). The individual = symbol assigns value while the == symbol checks if two things are equal.Let's try a little something. Look at the following irb session.irb :001 > a = 4 => 4 irb :002 > b = a => 4 irb :003 > a = 7 => 7 What is the value of b at this point? Take your best guess and then type this session into irb to find out.You'll notice that the value of b remains 4, while a was re-assigned to 7. This shows that variables point to values in memory, and are not deeply linked to each other. If this is confusing, don't worry, we'll have plenty of exercises for you to complete that will make this information clear and obvious. And when in doubt, always try it out in irb.Getting Data from a UserUp until now, you've only been able to assign data to variables from within the program. However, in the wild, you'll want other people to be able to interact with your programs in interesting ways. In order to do that, we have to allow the user to store information in variables as well. Then, we can decide what we'd like to do with that data.One way to get information from the user is to call the gets method. gets stands for "get string", and is a lot of fun. When you use it, the program waits for the user to 1) type in information and 2) press the enter key. Let's try it out. Type these examples in irb to get the feel and play around with them for a bit if you'd like to.irb :001 > name = gets Bob => "Bob\n" After the code, name = gets, the computer waited for us to type in some information. We typed "Bob" and then pressed enter and the program returned "Bob\n". The \n at the end is the "newline" character and represents the enter key. But we don't want that as part of our string. We'll use chomp chained to getsto get rid of that - you can put .chomp after any string to remove the carriage return characters at the end.irb :001 > name = gets.chomp Bob => "Bob" There we go! That's much prettier. Now we can use the name variable as we so please.irb :001 > name = gets.chomp Bob => "Bob" irb :002 > name + ' is super great!' => "Bob is super great!" Variable ScopeA variable's scope determines where in a program a variable is available for use. A variable's scope is defined by where the variable is initialized or created. In Ruby, variable scope is defined by a block. A block is a piece of code following a method invocation, usually delimited by either curly braces {} or do/end. Be aware that not all do/end pairs imply a block*.Now that you have an idea of what constitutes a variable's scope, one rule that we want you to remember is this:Inner scope can access variables initialized in an outer scope, but not vice versa.Looking at some code will make this clearer. Let's say we have a file called scope.rb.# scope.rb a = 5 # variable is initialized in the outer scope 3.times do |n| # method invocation with a block a = 3 # is a accessible here, in an inner scope? end puts a What is the value of a when it is printed to the screen? Try it out.The value of a is 3. This is because a is available to the inner scope created by 3.times do ... end, which allowed the code to re-assign the value of a. In fact, it re-assigned it three times to 3. Let's try something else. We'll modify the same piece of code.# scope.rb a = 5 3.times do |n| # method invocation with a block a = 3 b = 5 # b is initialized in the inner scope end puts a puts b # is b accessible here, in the outer scope? What result did you get when running that program? You should have gotten an error to the tune of:scope.rb:11:in `<main>': undefined local variable or method `b' for main:Object (NameError) This is because the variable b is not available outside of the method invocation with a block where it is initialized. When we call puts b it is not available within that outer scope.* Note: the key distinguishing factor for deciding whether code delimited by {} or do/end is considered a block (and thereby creates a new scope for variables), is seeing if the {} or do/end immediately follows a method invocation. For example:arr = [1, 2, 3] for i in arr do a = 5 # a is initialized here end puts a # is it accessible here? The answer is yes. The reason is because the code did not create a new inner scope, since for is part of Ruby language and not a method invocation. When we use each, times and other method invocations, followed by {} or do/end, that's when a new block is created.Types of VariablesBefore we move on, you should be aware that there are five types of variables. Constants, global variables, class variables, instance variables, and local variables. While you should not worry too much about these topics in depth yet, here is a brief description of each.

Mathematics in Architecture

A PowerPoint presentation depicting how maths had been used during ancient times in Architecture.

Hyperbolic functions

Contents covered in this pdf regarding hyperbolic functions.In this pdf there is detailed concepts and problems related to the topic.


Contents covered in this ppt regarding Integration.In this ppt there is detailed explanation with programs.