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Let's see exponential distribution through one example. Let's assume that I get two calls every hour. So, I want to know what is the probability of me getting a call in the next hour? In this particular case, if you will see, every hour I'm getting two phone calls. Which means that there is a probability of getting one phone call every half an hour. I know this. So, here I can define my lambda.
What would be my lambda in this particular case? 1/2, (one by two) which means 0.5 would be my lambda’s value. Lambda, we had seen, it was our rate, that In which rate are any of my events happening, that we were defining with lambda. Now I have lambda’s value. The x that I can see in my formula. To define that I will choose one interval, which is between 0 to 1. Why? 0 to 1 means then I want to know the probability of the phone call that I will get in the next one hour. So that will define my interval.
When we will put lambda’s value and x’s value in our probability density function. So, I will get one particular value over there. Which will be as follows: 0.393469. When we put lambda’s and x's value in our PDF of exponential distribution, then I will get one probability, the probability of me getting a phone call in the next hour, How much are the chances or what is the probability that I got to know from this exponential distribution. Now, if I see the exponential distribution’s mean and variance then that depends on the lambda only. Lambda is a very important parameter in exponential distribution, which defines what would be the mean? My mean’s formula is one upon lambda and my variance is defined by one upon lambda square.
Now we will see once how lambda affects my distribution. Here we have drawn two curves. Here one is for lambda equals to one, which we are seeing with a blue curve and lambda is equal to two which we can see with a pink curve. In both these curves, you must be seeing that my curve of lambda is equal to one. It is on the lower end, which means it is a more flattened curve. And when my lambda is two, which means it is a higher lambda then my curve is a little steep. Which means it is a little of the upper side and it is dropping more. So, with this we get to know that lambda highly affects my curve. If my lambda’s value is higher, which means if my rate is greater then my curve will drop very quickly and easily. But if my rate is lower, then my curve would be more flatten. In that way, we can define how lambda affects my rate. We will see this with one more example in which ways lambda affects, lambda tells us that in our exponential function, how quickly decay will happen, which means how easily looking at the curves, we can define the rate's value, and we can tell how much more my decay is happening. So, if we see this example, where we have drawn three curves, for three different rates,
First I have drawn a red curve, whose rate is equal to 1, in that case if you see that my curve is on the top, which means it defines that there is no decay. If my curve is closer to one, which means if I see the value rate is equal to 0.5, so this value is near to 0. This shows that our decay is slow. Which means my curve will keep flattening. If my value is closer to 0, let’s say 0.25, which we are seeing in the green curve. It is my completely steep decay, which means it is a completely flatten curve.
So, exponential distribution basically tells me how easily the decay will happen. And lambda plays a very important role. By defining lambda in the probability distribution curve, We can get to know the expected time of any event happening.
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