Uniform, Normal, and Exponential Distributions A probability distribution defines the probability of a random variable taking specific values within a certai...
A probability distribution defines the probability of a random variable taking specific values within a certain range. These distributions play a crucial role in probability theory and are used in various engineering applications.
Uniform Distribution:
Imagine a continuous, infinite line segment with no definite length. This line segment represents the possible values the random variable can take.
The probability density function (pdf) of the uniform distribution is constant within the interval [a, b], where a and b are the endpoints of the line segment.
The probability of a specific value within this range is determined by the length of the interval corresponding to that specific value.
Examples: throwing a coin in the air, measuring the length of a material, or determining the price of a stock at a specific time.
Normal Distribution (Gaussian Distribution):
This is the most widely used continuous probability distribution in engineering and probability theory.
It is characterized by a bell-shaped curve that is centered at the mean and symmetric on both sides.
The probability density function (pdf) of the normal distribution is also constant within the interval [a, b], where a and b are the parameters of the distribution.
Parameters: mean (μ) and standard deviation (σ). The mean represents the center of the curve, and the standard deviation measures how spread out the distribution is.
Examples: measuring the height of a person, the weight of an object, or the time taken to fail a test.
Exponential Distribution:
This distribution describes the time between events in a stochastic process.
The probability density function (pdf) of the exponential distribution is given by: