Welcome to lecture 31 about discrete probability distributions. In this lecture, we will understand what is probability distributions as well as some of the discrete probability distribution such as binomial probability distributions and Poisson probability distributions. In the previous lecture, we talked about various graphical methods of describing a set of data. They were part of descriptive statistics. However, the inferential statistics deals with inferring about population parameters based on descriptive parameters so, As mean and standard deviation. probability distribution plays a major role in inferential statistics.

In order to understand inference and probability distribution. Let us consider some of the situations. a salesperson knows that 20% of customers mature and the first call itself he need to call eight customers on a particular day. He would like to know the probability maturing to four or six customers in the first call itself. In another situation and manager at radical Silverstein would like to know the probability of arriving tended customers in five hours. Take one more situation where a screw manufacturing process manager would like to know the percentage of screws falling within a particular range of diameter etc, different and complex situations can we help them in predicting the probabilities?

Over centuries statisticians developed various probability distributions to predict the chance of occurrence of some event of interest. They are called probability distributions, which are either table formulae paragraph that describes the value of variables and the probability associated with these values. Basically, probability distributions are of two types, namely discrete probability distribution for discrete data types, most commonly used discrete probability distributions are by nominal distributions and Poisson distributions. Similarly, for continuous data types, we use the continuous probability distributions. They are namely normal distributions students t distributions Chi square distributions, f distributions, etc. In this lecture, we will discuss about discrete probability distribution.

We will discuss about the continuous probability distribution in the next lecture. The conditions for using binomial distribution are it consists of a fixed number of trials. And for each trial there are two possible outcomes can label the one outcome as success and other as failure. probability of success is B and probability of failure is one minus p The symbology relevant to binomial distribution our sample size, our number of trials, denoted as n, number of successes in the sample, denoted as x. Probability of a success for each trial, denoted as P. key assumptions are each trial is independent. Each trial can result only two outcomes, success or failure. The equation for binomial formula is as you can see on the screen let us take an example a sample of six products selected from a batch With non conforming percentage known as 15% can we find the probability that there are exactly two non conforming units in the sample?

We can solve this by using binomial distribution. Care, n is equals to six, and x is equals to two. So we need to find p of x is equals to two, which is equal to six factorial into 0.15 square into one minus 0.15 old square divided by two factorial into four factorial the problem Probability comes as 0.1762 it means probability of getting exact two nonconformities from this sample is equal to 17.62% it is tedious to calculate, I will show how to calculate the same in excel at the last of this lecture. But now, let me discuss about the next probability distribution for discrete data. Why isn't probability distribution is very useful probability distribution in six sigma, like binomial random variable of interest or is also named us Success of an event the difference between binomial and Poisson our random variable in binomial is success and a set of trial and random variable in poison is success in an interval of time or specified region of space.

The symbology relevant to Poisson distribution, our number of success in an interval denoted as x, mean number of successes, denoted as mew in an interval. The equation for a Poisson distribution is as you can see on the screen let us learn with an example. A book publisher knows that there typographical error is Poisson distributed with mean as 1.5 errors for 100 pages. Now, can he calculate the probability of typographical error to be zero and 400 pages of new book published recently, we can solve this with Poisson distribution table. Given our mean is equals to 1.5 errors per page. That means, for 400 pages, it will be six x is equals to zero.

So, we have to find out the probability of getting zero error. In other words, P of zero equals to E raised to minus six into mew raised to zero divided by zero factorial, this is equals to 0.00 to four. This means the probability of getting zero typographical error in the newly published book is 0.24% only as I have told that the calculation for both distribution are tedious excel sheet could be used for finding these distributions. Let us see how for binomial probability, we need to know probability success be number of success, x and number of trials and formula for Excel is binomial test into x and p, true or false. The binomial probability can be calculated in Excel for the above, discussed example, as, the binomial probability can be calculated in Excel for the above discussed example as, select a cell and type equals to binomial list in brackets, two, comma six comma 0.15 comma false, close the bracket and enter.

We will get the same answer as 0.176 Similarly, for boysson we need to know number of successes in an interval x the number of successes new formula for Excel is equals to poison open the bracket x comma mew comma true or false and close the bracket. The example for poison can be calculated in Excel as select a cell and type equals to poison. Open the bracket zero comma six comma false and close the bracket and enter. We will get the same answer As 0.00247 and of the lecture, we can proceed to learn about various continuous probability distribution in the next lecture. Thank you.