Poisson and Normal - evaluation of statistical parameters for these three distributions

Definition

A probability distribution is a mathematical function that assigns probabilities to possible values of a random variable. The Poisson distribution describes the number of events occurring in a fixed interval of time, area, or space when the events happen independently and at a constant average rate. The Normal distribution describes a continuous variable that is symmetrically distributed around its mean in a bell-shaped curve. The evaluation of statistical parameters means determining and interpreting measures such as mean, variance, and standard deviation to describe these distributions quantitatively.

For the commonly studied trio of distributions in this unit—typically Poisson, Binomial, and Normal—parameter evaluation means:

• identifying the key parameter(s) that define each distribution,
• computing the mean and variance,
• and understanding how the distributions behave under different conditions.

Main Content

1. Poisson Distribution

Meaning and use

• The Poisson distribution models the number of times an event occurs in a fixed interval when events occur independently and with a constant average rate. Examples include number of phone calls per hour, accidents at a traffic junction per day, or defects per meter of fabric.

Statistical parameters

If $X \sim \text{Poisson}(\lambda)$, then:

• Mean $= \lambda$
• Variance $= \lambda$
• Standard deviation $= \sqrt{\lambda}$
  Here, $\lambda$ is the average number of events in the interval and is the only parameter of the distribution.

Key properties and interpretation

The Poisson distribution is discrete, with values $0,1,2,3,\dots$. It is skewed to the right when $\lambda$ is small, and becomes more symmetric as $\lambda$ increases. A useful property is that the mean and variance are equal, which is a major identifying feature in parameter evaluation.
Formula: $$P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!}, \quad x=0,1,2,\dots$$ Example: If the average number of emails received per hour is 4, then $\lambda=4$. The expected number of emails in an hour is 4, and the variance is also 4.

2. Normal Distribution

Meaning and use

• The Normal distribution describes continuous data that naturally cluster around a central value. It is common in measurements such as heights, test scores, measurement errors, blood pressure, and manufacturing tolerances.

Statistical parameters

If $X \sim N(\mu,\sigma^2)$, then:

• Mean $= \mu$
• Variance $= \sigma^2$
• Standard deviation $= \sigma$
  The parameter $\mu$ determines the center of the curve, while $\sigma$ controls the spread.

Key properties and interpretation

The curve is symmetric about the mean, so the mean, median, and mode are equal. About 68% of observations lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This is called the empirical rule.
Density function: $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}$$ Example: If the average height of students in a class is 170 cm with standard deviation 6 cm, then the distribution can be written as $N(170,36)$. The spread indicates how much individual heights differ from the average.

3. Binomial Distribution

Meaning and use

• The Binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two outcomes: success or failure. Examples include number of heads in coin tosses, number of defective items in a sample, or number of students passing an exam.

Statistical parameters

If $X \sim B(n,p)$, then:

• Mean $= np$
• Variance $= npq$, where $q=1-p$
• Standard deviation $= \sqrt{npq}$
  Here, $n$ is the number of trials, $p$ is the probability of success, and $q$ is the probability of failure.

Key properties and interpretation

The Binomial distribution is discrete and depends on two parameters. It is symmetric when $p=0.5$, and becomes skewed when $p$ is closer to 0 or 1. It is also important because, under certain conditions, the Poisson distribution can be used as an approximation to the Binomial distribution, and the Normal distribution can also approximate the Binomial when sample size is large.
Formula: $$P(X=x)=\binom{n}{x}p^xq^{n-x}, \quad x=0,1,2,\dots,n$$ Example: If a coin is tossed 10 times and the probability of heads is 0.5, then the mean number of heads is $10 \times 0.5 = 5$, and the variance is $10 \times 0.5 \times 0.5 = 2.5$.

Working / Process

1. Identify the type of data and distribution

• Determine whether the variable is discrete count data, continuous measurement data, or a sequence of success/failure trials.
• Use Poisson for counts in fixed intervals, Normal for continuous bell-shaped data, and Binomial for repeated binary trials.

2. Calculate the statistical parameters

• For Poisson, estimate $\lambda$ from the average count.
• For Normal, determine $\mu$ and $\sigma$ from the data.
• For Binomial, identify $n$ and $p$, then compute $np$, $npq$, and $\sqrt{npq}$.

3. Evaluate and interpret the distribution

• Compare mean and variance to understand spread.
• Check shape characteristics such as symmetry or skewness.
• Use the parameters to make probability predictions, comparisons, or approximations.

Advantages / Applications

• Helps in modeling real-world random events such as arrivals, defects, and counts in time or space.
• Supports prediction and decision-making in engineering, business, healthcare, and science.
• Provides a foundation for approximation methods, such as using Normal to approximate Binomial or Poisson under suitable conditions.

Summary

• Poisson models event counts, Normal models continuous measurements, and Binomial models repeated success-failure trials.
• The main statistical parameters are mean, variance, and standard deviation.
• Poisson has one parameter $\lambda$, Normal has $\mu$ and $\sigma^2$, and Binomial has $n$ and $p$.
• Important terms to remember: mean, variance, standard deviation, Poisson parameter $\lambda$, Normal mean $\mu$, Normal standard deviation $\sigma$, Binomial success probability $p$