# Common Probability Distributions

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##### 1. Normal Distribution:

The Normal Distribution equation is given by:

$$f(x) = \frac{1}{\sigma\sqrt{2 \pi }} e^{ -\frac{(x - \mu)^2}{2 \sigma^2}}$$

where μ is the mean, σ is the standard deviation, and x is the random variable. This equation describes the probability density function of the Normal Distribution, which is a continuous probability distribution that is symmetrical around the mean. It is often used to model real-world phenomena that are approximately normally distributed, such as height, weight, and IQ scores.

Where:

π ≈ 3.141

e ≈ 2.718

##### 2. Binomial Distribution:

Where:

p = Probability of success

q = Probability of failure =  1−p
n = Number of trials

P(x) = Probability of x successes in n trials

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials. The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success (p) in each trial. The formula for the binomial distribution is as follows:

$${P(x)} = ^{n}{C_x}.{p^x}.{q^{n-x}}$$

where:

• P(X = x) is the probability of x successes in the given number of trials
• n choose x is the binomial coefficient, which is the number of ways x successes can occur in n trials
• p^x is the probability of x successes in n trials
• q^(n-x) or (1-p)^(n-x) is the probability of (n-x) failures in n trials

For example, if we have a binomial distribution with 10 trials and a probability of success of 0.5 in each trial, the probability of getting exactly 5 successes in those 10 trials would be:

P(X = 5) = (10 choose 5) * 0.5^5 * 0.5^(10-5)

= 252 * 0.03125 * 0.03125

= 0.207

Thus, the probability of getting exactly 5 successes in 10 trials with a probability of success of 0.5 in each trial is 0.207.

##### 3. Poisson Distribution:

The Poisson distribution is a statistical distribution that can be used to model the number of occurrences of a given event in a fixed period of time, space, or area. It is often used to model the number of times a specific event occurs in a given period of time, such as the number of cars that pass through a particular intersection in an hour, or the number of calls to a customer service center in a day.

The formula for the Poisson distribution is:

$${P(x)} = {e^{-λ}}.\frac{λ^x}{x!}$$

where:

• P(x) is the probability of x occurrences
• λ is the average number of occurrences in the given period of time, space, or area
• e is the mathematical constant approximately equal to 2.71828
• x! is the factorial of x (the product of all positive integers less than or equal to x)

For example, if the average number of cars that pass through a particular intersection in an hour is 10, the probability of exactly 15 cars passing through that intersection in an hour is:

P(15) = (e^(-10) * 10^15) / 15! = (2.71828^(-10) * 10^15) / 15! = 0.000045

In other words, the probability of 15 cars passing through the intersection in an hour is very low, according to the Poisson distribution.

Posted on December 15, 2018 by  Quality Gurus

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