2. Key Discrete and Continuous Distributions

The formula for the Binomial Distribution is:

P(X=r) = \binom{n}{r} p^r (1-p)^{n-r}

The Binomial Distribution describes the number of successes in a fixed number of independent trials of a binary experiment. Each trial results in a success with probability 'p' and a failure with probability '1-p'. The formula calculates the probability of observing 'r' successes in 'n' trials.

• $\binom{n}{r}$ is the combination function, which gives the number of ways to choose 'r' successes from 'n' trials.
• $p^r$ represents the probability of having 'r' successes.
• $(1-p)^{n-r}$ depicts the probability of 'n-r' failures.

Think of flipping a coin 10 times, where the coin has a 60% chance of landing heads (success). We can use the Binomial Distribution to find the probability of getting a certain number of heads (e.g., 7 heads).

The formula for the Poisson Distribution is:

P(X=r) = \frac{\lambda^r e^{-\lambda}}{r!}

The Poisson Distribution is used for modeling the number of events that occur within a fixed interval of time or space, given that these events happen with a known constant mean rate ($\lambda$) and are independent of the time since the last event.

• $\lambda^r$ is the rate of occurrence raised to the number of occurrences (r).
• $e^{-\lambda}$ accounts for the decay of probability.
• $r!$ is the factorial of 'r', used to account for all possible arrangements of the 'r' successes.

Imagine you are at a call center, and on average, you receive 3 calls every hour. The Poisson Distribution can help determine the probability of receiving exactly 5 calls within that hour.

The formula for the Normal Distribution is:

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

The Normal Distribution, often referred to as the bell curve, is essential in statistics and represents real-valued random variables whose distributions are not known. It is characterized by its mean ($\mu$) and standard deviation ($\sigma$).

• $f(x)$ gives the probability density function at point 'x'.
• The terms within the exponent determine how spread out the distribution is. Many statistical methods rely on the normality assumption thanks to the Central Limit Theorem.

Consider the heights of adult men in a specific country. If we collect enough data, we will find that most men are around a certain height (the mean), with fewer men being very tall or very short. This distribution of heights resembles the shape of a bell curve.

For the Binomial, Poisson, and Normal distributions, the following parameters are evaluated:

• Mean
• Variance
• Standard Deviation

The evaluation of parameters for distributions is vital for understanding their characteristics:

1. Mean: Represents the average outcome expected from the distribution.
2. Variance: Measures how spread out the values are from the mean. The higher the variance, the more spread out the distribution.
3. Standard Deviation: The square root of the variance, giving a measure of spread in the same units as the data. Each distribution has its specific formulas for calculating these parameters based on its characteristics.

If you think of a class's exam scores, the mean gives you a sense of the class's overall performance, variance tells you how consistent the scores are, and standard deviation tells you how much the scores deviate from the average. Various distributions will exhibit different types of spreads based on their applications.

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