Normal distribution, exponential & Poisson distributions

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A normal distribution is on that occurs naturally in many situations, it is called the bell curve distribution. In a normal distribution, the data tends to be around a central value with no bias to left or right and gets close to normal distribution or a bell curve(1). The normal distribution gets its importance from the central limit theorem. According to the central limit theorem the averages of sample observations of random variables independently drawn from independent distributions converge in distribution to the normal. When observations are sufficiently large, the distributions converge to the normal(2). Every normal distribution is a version of the standard normal distribution whose domain has been stretched by its standard deviation and then translated by the mean value.

The exponential distribution is used to show the time elapsed between events. If the waiting time is unknown, it is appropriate to think of an event as having an exponential distribution. The time X we need to wait before an event occurs has an exponential distribution. The time X that needs to be waited before an event occurs has an exponential distribution if the probability that the even occurs during a certain time interval is proportional to the length of the time interval(3). The mean of an exponentially distributed random variable X with rate parameter lambda is given by E(X) = 1/ lambda = beta. For example, if a store receives customers at 3 per hour, then the storekeeper can expect to wait 20 minutes for every customer. The variance of X is given by var(X) = 1/( lambda) squared = Beta squared. In an exponential distribution the standard deviation is equal to the mean.

The Poisson random variable is the ...