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# Theoretical Distribution

In the real world, we rarely come across experiments with single outcomes like heads or tails. Generally, we do the experiment as a set of events and carry it for n number of times which give us a collection of outcomes which we can represent in the form of theoretical distribution. By theoretical distribution, we take mean of a frequency distribution, which we obtain in relation to a random variable by some mathematical model.

## Theoretical Distribution

A random exponent is assumed as a model for theoretical distribution, and the probabilities are given by a function of the random variable is called probability function.

For example, if we toss a fair coin, the probability of getting a head is $$\frac{1}{2}$$. If we toss it for 50 times, the probability of getting a head is 25. We call this as the theoretical or expected frequency of the heads. But actually, by tossing a coin, we may get 25, 30 or 35 heads which we call as the observed frequency.

Thus, the observed frequency and the expected frequency may equal or may differ from each other due to fluctuation in the experiment.

### Types of Theoretical Distribution

1. Binomial Distribution
2. Poisson distribution
3. Normal distribution or Expected Frequency distribution

#### Binomial Distribution:

The prefix ‘Bi’ means two or twice. A binomial distribution can be understood as the probability of a trail with two and only two outcomes. It is a type of distribution that has two different outcomes namely, ‘success’ and ‘failure’. Also, it is applicable to discrete random variables only.

Thus, the binomial distribution summarized the number of trials, survey or experiment conducted. It is very useful when each outcome has an equal chance of attaining a particular value. The binomial distribution has some assumptions which show that there is only one outcome and this outcome has an equal chance of occurrence.

The three different criteria of binomial distributions are:

1. The number of the trial or the experiment must be fixed.
2. Every trial is independentNone of your trials should affect the possibility of the next trial.
3. The probability always stays the same and equal. The probability of success may be equal for more than one trial.

#### Source: freepik.com

#### Poisson Distribution :

The Poisson Distribution is a theoretical discrete probability distribution that is very useful in situations where the events occur in a continuous manner. Poisson Distribution is utilized to determine the probability of exactly x0 number of successes taking place in unit time. Let us now discuss the Poisson Model.

At first, we divide the time into n number of small intervals, such that → ∞ and denote the probability of success, as we have already divided the time into infinitely small intervals so p → 0. So the result must be that in that condition is n x p = λ (a finite constant).

#### Normal Distribution :

The Normal Distribution defines a probability density function f(x) for the continuous random variable X considered in the system. The random variables which follow the normal distribution are ones whose values can assume any known value in a given range.

We can hence extend the range to – ∞ to  + ∞ . Continuous Variables are such random variables and thus, the Normal Distribution gives you the probability of your value being in a particular range for a given trial.  The normal distribution is very important in the statistical analysis due to the central limit theorem.

The theorem states that any distribution becomes normally distributed when the number of variables is sufficiently largeFor instance, the binomial distribution tends to change into the normal distribution with mean and variance.

## Solved Example on Theoretical Distribution

Explain the properties of Poisson Model and Normal Distribution.

#### Properties of Poisson Model :

• The event or success is something that can be counted in whole numbers.
• The probability of having success in a time interval is independent of any of its previous occurrence.
• The average frequency of successes in a unit time interval is known.
• The probability of more than one success in unit time is very low.

#### Properties of Normal Distribution :

• Its shape is symmetric.
• The mean and median are the same and lie in the middle of the distribution
• Its standard deviation measures the distance on the distribution from the mean to the inflection point (the place where the curve changes from an “upside-down-bowl” shape to a “right-side-up-bowl” shape).
• Because of its unique bell shape, probabilities for the normal distribution follow the Empirical Rule, which says the following:
• About 68 percent of its values lie within one standard deviation of the mean. To find this range, take the value of the standard deviation, then find the mean plus this amount, and the mean minus this amount.
• About 95 percent of its values lie within two standard deviations of the mean.
• Almost all of its values lie within three standard deviations of the mean.
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Hood

In problem two, the sum of standard deviations should be 4.4 not 3.16.

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