Binomial Distribution

    The binomial distribution is a sequence of experiments that are repeated a fixed number of times to measure the probability of obtaining a specific number of successes. Essentially, it is the culmination of a fixed number of repetitive independent and identical bernoulli trials.

For a random variable, $X$, a binomial distribution with a probability of observing exactly $x$ successes, $P(X=x)$ can be denoted as:

$$P(X=x) = \binom{n}{x} p^x (1-p)^{n-x} \quad \forall x \in {0,1,2,\dots,n}$$

There are other noteworthy characteristics of the binomial distribution, as aforementioned, for a random variable X which follows a binomial distribution:

1. $$E(X) = np$$
2. $$Var(X) = np(1-p)$$

Properties of a Binomial Distribution

    For an experiment to be modelled utilizing a binomial distribution, it must meet at least all the following properties:

1. The individual trials must end in either success or failure.
2. Each individual trial must be independent of each other, i.e., the outcome of one trial does not afffect another.
3. There is a fixed number of trials, $n$.
4. The probability of sucess, $p$, must be the same for each trial.

Binomial Distribution: Approximations

    The formula for the binomial distribution can become quite cumbersome for large values of $n$. Hence, there are suitable approximations which can be made to account for this:

Normal Approximation to Binomial

    The normal distribution can be utilized to approximate the discrete binomial distribution. It follows for a discrete random variable, $X$, modelled binomially with parameters, $n$ and $p$, such that $np \ge 5$ and $n(1-p) \ge 5$ then:

$$X \sim \mathcal{N}(np, npq), \quad \text{approximately.}$$

    It is crucial to note that we are utilizing a continuous distribution to approximate a discrete distribution, therefore, the random variable $X$ must undergo continuity corrections to account for this discreprancy, ensuring the probability mass is accurately captured:

1. $P(X > x) \rightarrow P(X > x+0.5)$
2. $P(X \ge x) \rightarrow P(X > x-0.5)$
3. $P(X < x) \rightarrow P(x < x-0.5)$
4. $P(X \le x) \rightarrow P(X < x+0.5)$

Poisson Aproximation to Binomial

    The poisson distribution can be utilized to approximate the discrete binomial distribution. It follows for a discrete random variable, $X$, modelled binomially with parameters, $n$ and $p$, and for large values of $n$, such that, $n > 50$ and $p \le 0.1$, then:

$$X \sim \mathrm{Po}(\lambda)$$

where, $\lambda = np$

    Once again, it is crucial to note that the poisson distribution is a discrete distribution akin to the binomial distribution, therefore, no continuity corrections are necessary to accurately capture the probability mass.

Questions

1. Rahul operates an automated manufacturing company which focuses on making Christmas ornaments. Every 5 ornaments, a defective ornament is produced. The machine produces a total of 50 ornaments in a day. i. What is the probability of an ornament being defective? (Ans: $0.2$) ii. What distribution can be utilized to model this scenario and state the parameters? (Ans: $X \sim \text{Bin}(50, 0.2)$) iii. What is the probability that there are five defective ornaments? (Ans: $0.0295$).

2. A newspaper company buys a shipment of 500 pens with each pen having a $15%$ chance of being broken upon arrival. i. What distribution can be utilized to model this scenario and state the parameters? (Ans: $X \sim \text{Bin}(500,0.15)$) ii. Using a normal approximation, stating the requirements, find that the probability that there are at MOST 75 broken pens. (Ans: $0.525 \text { (to 3 s.f.)}$).
   
3. A warden began collecting data on $2,000$ cars passing on a highway within a day. He discovered that there is a $1%$ chance that a car being involved in an accident. i. Using a binomial distribution, model this scenario and state the parameters. (Ans: $X \sim \text{Bin}(2000, 0.01)$) ii. Utilizing a poisson approximation, find the probability that there are at LEAST 25 car accidents in a day. (Ans: $0.11$)