Explain normal distribution and exponential distribution respectively in detail giving all the assumptions and relevant examples
1. Normal Distribution:
Normal distribution, also known as Gaussian distribution, is a probability distribution that is symmetric and bell-shaped. It is widely used in numerous fields due to its convenient mathematical properties and many natural phenomena follow this distribution. Normal distribution is defined by two parameters: mean (μ) and standard deviation (σ).
Assumptions:
- Data follows a unimodal, symmetric, and bell-shaped pattern.
- Data values are continuous and can take any real value.
- Each data point is independent and not affected by other data points.
- The mean, variance, and standard deviation of the distribution are finite and well-defined.
Examples:
i) Heights of Adult Males: The heights of adult males follow a normal distribution. The mean height is usually around 5 feet 9 inches, and the standard deviation is typically around 3 inches. The distribution of heights forms a bell-shaped curve where most men fall near the mean height, and fewer individuals are found towards the extremes (very tall or very short).
ii) IQ Scores: Intelligence quotient (IQ) scores are often assumed to follow a normal distribution. The mean IQ score is set to 100, and the standard deviation is set to 15. The majority of people have IQ scores around the mean, with fewer individuals having scores that deviate significantly higher or lower.
2. Exponential Distribution:
Exponential distribution is a probability distribution that describes the time between events in a Poisson process. It is commonly used in reliability analysis and queuing theory, where it models the time between failures or arrivals. Exponential distribution is characterized by a single parameter, λ (lambda), which represents the rate at which events occur.
Assumptions:
- Events occur independently and at a constant average rate.
- Time between events follows an exponential distribution.
- The probability of multiple events occurring in a given interval is negligible.
Examples:
i) Waiting Times: Suppose you are waiting at a bus stop, and buses arrive according to an average rate of one bus every 15 minutes. The waiting time until the next bus arrives can be modeled using an exponential distribution with a rate parameter of λ = 1/15.
ii) Machine Failure: The lifetime of a machine before it fails can often be modeled using an exponential distribution. For example, the time it takes for a light bulb to fail can be assumed to follow an exponential distribution with an average failure rate of λ.