Gamma Distributio n by Dr Mahamood Usman Khan Ph D IIT ISM Dhanbad M Phil M Sc AMU Gamma Distribution Two Parameter If X is a continuous random variable Then the probability density function pdf of gamma distribution is given as follows xf x 1 e x x 0 0 1 1 parameter scale parameter 1 shape Mean Variance 2 Moment Generating Function M t X 1 t 1 Gamma Distribution Two Parameter xf x 1 e x x 0 0 2 1 shape parameter rate parameter Another form of pdf Mean Variance 2 Generating Function Moment t M X t Behavior of Gamma pdf Graphical View Scale is fixed Shape is fixed Standard Gamma Distribution If we put 1 in 1 1 or 1 2 then we get the standard gamma distribution given as This is also known as the one parameter gamma distribution with shape parameter The mean and variance of this gamma distribution is same and equal to Incomplete Gamma Distribution When to Use Gamma Distribution Gamma Distribution is a Continuous Probability Distribution that is widely used in different fields of science to model continuous variables that are always positive and have skewed distributions It occurs naturally in the processes where the waiting times between events are relevant Use of Gamma distribution in Real Life Problems pollutant concentrations securities analysis The gamma distribution is applied in reliability engineering for analyzing system lifetimes and time until failure It finds use in insurance and risk analysis for assessing claim frequencies and Also used in environmental sciences for modeling variables like rainfall and In finance and economics the gamma distribution aids in option pricing and risk In healthcare and bio statistics it is utilized for studying patient waiting times cancer rates and drug dosage responses Additionally the gamma distribution is employed in image and signal processing for noise modeling and restoration In traffic engineering it s used for analyzing traffic flows and delays in transportation networks Example 1 The lifetime of a certain type of electronic component follows a gamma distribution It is known that the average lifetime of these components is 500 hours and the variance is 100 000 hours squared Find the probability that a randomly selected component will last at least 700 hours Solution Example 2 Suppose that when a transistor of a certain type is subjected to an accelerated life test the lifetime X in weeks has a gamma distribution with mean 24 weeks and standard deviation 12 weeks a What is the probability that a transistor will last between 12 and 24 weeks b What is the probability that a transistor will last at most 24 weeks Solution Properties of Gamma Distribution ThankYou