** Binomial vs Normal Distribution **

Probability distributions of random variables play an important role in the field of statistics. Out of those probability distributions, binomial distribution and normal distribution are two of the most commonly occurring ones in the real life.

**What is binomial distribution?**

Binomial distribution is the probability distribution corresponding to the random variable *X, *which is the number of successes of a *finite sequence* of independent yes/no experiments each of which has a probability of success *p*. From the definition of *X,* it is evident that it is a discrete random variable; therefore, binomial distribution is discrete too.

The distribution is denoted as *X *~*B*(*n*,*p*) where *n *is the number of experiments and p is the probability of success. According to probability theory, we can deduce that *B*(*n*,*p*) follows the probability mass function [latex] B(n,p)\\sim \\binom{n}{k} p^{k} (1-p)^{(n-k)}, k= 0, 1, 2, …n [/latex]. From this equation, it can be further deduced that the expected value of *X*, E(*X*) = *np *and the variance of *X*, V(*X*) = *np*(1-*p*).

For example, consider a random experiment of tossing a coin 3 times. Define success as obtaining H, failure as obtaining T and the random variable *X *as the number of successes in the experiment. Then *X*~*B*(3, 0.5) and the probability mass function of *X* given by [latex] \\binom{3}{k} 0.5^{k} (0.5)^{(3-k)}, k= 0, 1, 2.[/latex]. Therefore, the probability of obtaining at least 2 H’s is P(*X ≥ *2) = P (*X *= 2 or *X *= 3) = P (*X *= 2) + P (*X *= 3) = ^{3}C_{2}(0.5^{2})(0.5^{1}) + ^{3}C_{3}(0.5^{3})(0.5^{0}) = 0.375 + 0.125 = 0.5.

**What is normal distribution?**

Normal distribution is the continuous probability distribution defined by the probability density function, [latex] N(\\mu , \\sigma)\\sim\\frac{1}{\\sqrt{2 \\pi \\sigma^{2}}} \\ e^{- \\frac{(x-\\mu)^{2}}{2 \\sigma^{2}}} [/latex]. The parameters [latex] \\mu and \\sigma [/latex] denote the mean and the standard deviation of the population of interest. When [latex] \\mu = 0 and \\sigma = 1 [/latex] the distribution is called the standard normal distribution.

This distribution is called normal since most of the natural phenomena follow the normal distribution. For, example the IQ of the human population is normally distributed. As seen from the graph it is unimodal, symmetric about the mean and bell shaped. The mean, mode, and median are coinciding. The area under the curve corresponds to the portion of the population, satisfying a given condition.

The portions of population in the interval [latex] (\\mu – \\sigma, \\mu + \\sigma) [/latex], [latex] (\\mu – 2 \\sigma , \\mu + 2 \\sigma) [/latex], [latex] (\\mu – 3 \\sigma , \\mu + 3 \\sigma) [/latex] are approximately 68.2%, 95.6% and 99.8% respectively.

**What is the difference between Binomial and Normal Distributions?**

- Binomial distribution is a discrete probability distribution whereas the normal distribution is a continuous one.
- The probability mass function of the binomial distribution is [latex]B(n,p)\\sim \\binom{n}{k} p^{k} (1-p)^{(n-k)} [/latex], whereas the probability density function of the normal distribution is [latex] N(\\mu, \\sigma)\\sim\\frac{1}{\\sqrt{2 \\pi \\sigma^{2}}} \\ e^{- \\frac{(x-\\mu)^{2}}{2 \\sigma^{2}}} [/latex]
- Binomial distribution is approximated with normal distribution under certain conditions but not the other way around.

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