Discrete Probability Distribution

We will start this chapter with discrete probability distribution.

The first probability distribution that we will see here is Bernoulli distribution.

Bernoulli distribution (short pause) is that type of distribution in which we perform the experiment only once and we generate one distribution.

In this probability distribution, we only have two outcomes, which we define with success or failure.

So, this means, if I say “p”, that defines my probability of success “1-p” or which we also call as Q.

It defines my probability of my failure.

Let's understand this with an example.

Suppose, you have flipped one coin, after flipping it, you have two possible outcomes with you.

Either head will come or tail will come, the probability of head can be success and the probability of tail can be a failure.

These we can see in different use cases.

When we play cricket, to decide who will bat first or who will be bowling first.

We toss a coin, in that particular scenario whoever wins that toss, that person decides who will play first.

In the same way, we take different scenarios.

If there is any question whose outcomes are only in yes or no that will be calculated in Bernoulli distribution.

Whether it will rain tomorrow or not, even this is an example of Bernoulli distribution.

Why? Because the event of rain can be called as my success.

The event of no rain can be called my failure.

So, any such distribution which has two possible outcomes that would be called my Bernoulli Distribution.

The distribution that I get after performing the event once is called the Bernoulli distribution.

Its probability mass function formula is as follows: p to the power x, one minus p to the power, one minus x, where my x can be any outcome, zero or one.

If I place the values in it, then the px value would be one minus p, when my x is zero and when my x is one then its value would be p.

Like we can see in this graph.

On the x axis I have taken a random variable and on the y axis I have taken the probability mass function, this graph defines probability of success which means when p is one, it defines then and when my random variable is zero, then my probability is one minus p.

It defines that.

Now, one important thing is the different trials that we take in Bernoulli distribution, we call them Bernoulli trials.

Bernoulli distribution is one important distribution which forms a base for the other type of distribution, which we will see in the coming chapters.

Success and failure are subjective events which means as and when our use case will change.

Accordingly, our success and failure would be defined.

Like in the case of tossing a coin, coming up head or tail that defines our success or failure.

Will it rain or not, in that particular case that would define my success or failure.

Now what would be the mean or variance of my Bernoulli distribution? Let's see that.

Mean of any Bernoulli distribution would be, e for x random variable is equal to one minus p multiplied by zero plus p multiplied by one.

It means that this is defined by p and the variance for x random variables, that is p multiplied by one minus p.

So, you remember this mean and variance.

Next, we will see binomial distribution.

Now, what is binomial distribution? When we perform any experiment n number of times and in those n number of times, we note all the probabilities.

So, the particular curve or graph that I get is called my binomial distribution.

So, basically my probability is…If in any n total trials I want to calculate r’s success probability.

So, far that there is a formula that I have, which means that my r here is my number of success if I've done n trials, in that particular case, my formula is, probability when the random variable x is equal to r is given by, ncr, p to the power r, one minus p to the power, n minus r.

You might be finding this formula difficult, but the ncr in it, we have seen in our primary classes that it is called the combination which gets elaborated and is defined as n factorial upon n minus r factorial multiplied by r factorial.

If we put different values into it, of n, r and p then we get all possible scenarios of probability.

In this, n is my total number of trials.

P is my probability of success in one trial, which means if I have tossed the coin once, the probability of getting a head or a tail, whatever success I have defined, that would be my p’s value and n means how many times I have tossed the coin that would define my n’s value.

Now, let's understand this with a particular example.

You might remember that we had played a game in our last module.

What was there in that game? I had five balls in my bag from which three were red balls and two were black balls.

In that we had told every person that you draw the ball four times.

Note its colour and keep the ball back in the bag.

Now you pay attention to this, the four times that we made every person draw the ball that would be called my total number of trials.

Which means my n’s value in that particular scenario becomes four.

And probability…suppose we believe that in one trial, the probability of getting the red ball which is p is 0.6.

So, if I have to find if r red balls are drawn, which means two red balls come, three red balls come, four red balls come, in that particular case.

All the probabilities, if I want to calculate without doing any experiment or without playing any game.

So how will I calculate that? We can calculate it through binomial distribution.

How? The formula that we just saw, which was px equals r, is given by ncr, p to the power r, one minus p to the power, n minus r, which means if I put n’s value as 4 and p's value as 0.6.

So, in the formulas, I will get these values, in the place of r if I have to find zero red balls.

In that particular case.

Instead of r I will put zero value.

So, if I'm calculating the value on px equals to zero, then that value comes as 4C0, 0.6 to the power of 0, 0.4.

to the power 4, which comes as 0.0256.

In the same way, if I calculate that value for x equals one, so the value that comes to me is 0.1536.

In the same way, we can find without playing the game that in different scenarios what is the probability of drawing red balls.

So, if I put x as 2, this means I can find the probability of drawing 2 red balls.

So, we have now understood what binomial distribution is, what are its examples, Now we will see properties of binomial distribution.

This means that we have got four properties of binomial distribution, which after executing them successfully will be called an example of binomial distribution.

First is, each trial is independent, which means we have seen in the case of probability that what are independent and dependent events.

This means that if my previous toss is not affecting the outcome of my current toss, which means if I have tossed a coin and the other coin that I have tossed, at other times, it is not affected by my first toss.

So, my trials will be called independent.

The second property is that the total number of trials should be fixed.

Third property is every trial should be binary, which means there can only be two possible outcomes of it, either success or failure.

Fourth property is that the probability of success in every trial should be the same.

 Now, let's see this with an example.

Suppose, if I have tossed one coin 20 times, I have to find out the probability that how many times tail will come in that particular case.

So, this would be my binomial distribution experiment.

In which ways? My total number of trials is 20, those are fixed.

The outcomes are two with me, either it can be head or tail and the probability in any event of tail, that is 0.5.

No matter how many times I toss it.

Plus, the event is independent, the trials are independent, which means not even one outcome is getting affected by the previous outcome.

So, this is very important that we should know all the properties of binomial distribution.

Now, let's see what is the meaning of the binomial distribution and what is the variance of it.

You might remember, what was Bernoulli distribution? In that we had mean p.

Since we have n number of events so, our mean becomes n multiplied by p.

Variance of my Bernoulli’s distribution was p multiplied by 1-p.

Since in binomial I have n number of events.

So, its variance becomes n multiplied by p multiplied by one minus p.

(short Pause) Now, let's come on to a very important topic.

What is the difference between Bernoulli and binomial Distribution? Since we have covered both the topics, we got to know that there are few similarities between them but this is a very important fact that Bernoulli distribution is a special case of Binomial distribution.

If I have a single trial with me, which means it is called a Bernoulli distribution when I have a single trial of events.

It is called binomial when I conduct the same experiment n number of times.

In Bernoulli distribution my mean is p, in binomial it is n multiplied by p.

Variance of Bernoulli distribution is p multiplied by one minus p and binomial’s variance is n multiplied by n multiplied by p multiplied by 1-p.

 If I take an example of Bernoulli, suppose I have to find out the probability of passing any exam.

Suppose that is 80%, probability of success and 20% is probability of failure which means, if I pass the class then I have 80% chance of passing and 20% chance of failing.

So, this is Bernoulli distribution.

Why? Because I have only two possible outcomes, either I can fail or I can pass.

Now suppose, in the same scenario, I have to find out the probability that from four out of five exams, what is the probability that I can pass or fail.

So, this would be an example of binomial distribution.

Why? Because I'm conducting an experiment to find four out of five probabilities.

So, n is 5 here, as they are the total number of outcomes.

4 here is r’s value.

The probability p of success is 80%.

Which means 0.8.

So, if I put the values in the formula then I will get different values.

So, here we have covered the difference between Bernoulli and binomial distribution.

Now, we will see our third and very important distribution, which comes in discrete probability distribution which is poisson distribution.

To understand poisson distribution let's take an example.

Let's take an example of customer care.

There we have to find out the number of calls that the customer care receives.

So, in this particular case, we can find out, we can have an estimate that every hour what can be the average number of calls the customer care can get.

But can you tell the exact number or exact time that this call will come at this time of an hour or only these many numbers of calls will come.

So, in this situation where the event is happening at a random point of time and at a random space, whenever this kind of situation arises at that time we use poisson distribution.

Our event is happening in any random point of time and space.

But our importance, we have to exactly find out what is the number of occurrences of that event happening.

Poisson distribution we apply when we have to count the number of times this event will occur.

Now, let's take different examples of it so that we can understand it easily.

Suppose, we have to find the number of emergency calls.

How many emergency calls come in a day at the hospital? This will be followed by my poisson distribution.

Why? Because we cannot say exactly how many calls can come, but we can give an estimate.

Let's take one more example.

The whole day, suppose, I am in one area.

I have got my house there.

In that area how many thefts happen in a day.

 In this particular case, how can we tell, what would be the total number of thefts or suppose you have to go to a parlour.

How many numbers of customers will come to the parlour in one hour or in particular city how many suicide rates will be reported.

Any such scenario where I do not know the exact number and I have to find it, at that time we apply Poisson distribution.

This particular distribution is basically a special case of binomial distribution where my n is reaching infinity and the expected number of successes, which means p is reaching zero, then my binomial distribution converts into a poisson distribution.

We can simply say that poisson becomes binomial if my n is large, and p is small.

The formula of poisson distribution is as follows p x equals to x is equals two e to the power minus lambda multiplied by lambda to the power x upon x factorial.

Here you must be seeing different terms.

So, x is called the average number of times, that is how many times my event has happened.

What is x? It is an event at any particular time.

E is Eulers number which is one constant value, 2.71818.

So, by using this particular formula we can choose our poisson's distribution.

 So, let's understand the Poisson distribution formula with one example.

If I have one data that in an hour in an hospital on an average two kids are born.

I have to determine in it, I have to find out whether exactly one kid should be born in that given hour or two or three kids.

So, in this situation, we can use poisson distribution.

How does it work? If we see this formula carefully, here I know E's value since it is a constant, Lambda here is two because my average value is two births per hour.

X’s value is the different value of the kids which we want to know.

If I want to find zero birth value, then I'll put zero instead of x.

If I want to find one birth value then instead of x I will put one, in the same way we can put more values.

Like if I have to find out that not even one kid was born in that hour.

Then in place of x, I will put 0.

If I put the values in the formula, lambda to the power x will mean two to the power zero multiplied by e to the power minus lambda, which will mean e to the power minus two, divided by zero factorial.

When I calculate this, I get one particular value 0.1353.

In the same way, if we put X's value as one, the  value comes 0.2707.

So, in the same way I have put two and three’s values.

And if I see its probability distribution curve.

On the x axis I have drawn birth and on the y axis I have taken probabilities so in this way, I will get one curve, which will be called a poisson distribution curve.

And we can easily find the exact number of values.

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