Verify that Var(X)=(t)/(λ^2) when X is a gamma random variable with parameters t and λ.

Name: Problem 19, Chapter 5: A First Course in Probability
Uploaded: 2022-01-13T02:55:23-08:00
Duration: 3 min 55 s
Channel: Amany Waheeb
Description: Verify that Var(X)=(t)/(λ^2) when X is a gamma random variable with parameters t and λ.

step 1 Hi guys in this problem we are given that x follows e gama distribution with parameter alpha and

Verify that Var(X)=(t)/(λ^2) when X is a gamma random...

Key Concepts

Gamma Distribution

The gamma distribution is a continuous probability distribution widely used to model waiting times and is characterized by two parameters: a shape parameter (often denoted as t) and a rate parameter (often denoted as ?). Its probability density function involves the gamma function and has the form f(x; t, ?) = (?^t / ?(t)) x^(t-1)e^(??x) for x > 0. This distribution generalizes the exponential distribution and plays a key role in various statistical applications.

Moments

Moments in probability theory are expected values of powers of the random variable and provide essential information about its distribution, such as the mean (first moment) and variance (derived from the first and second moments). For the gamma distribution, the mean is given by E[X] = t/? and the variance can be computed using the relation Var(X) = E[X^2] ? (E[X])^2, which eventually yields Var(X) = t/?².

Variance

Variance is a measure of how spread out a set of random values is around the mean. In the context of the gamma distribution, the variance is calculated using the second moment, and through properties of the gamma function and the definition of the distribution’s moments, it can be verified that Var(X) = t/?². This result encapsulates the distribution's dispersion relative to its parameters.

Gamma Function

The gamma function, denoted as ?(t), extends the factorial function to real and complex numbers, except for non-positive integers. It plays a crucial role in the gamma distribution, appearing in its probability density function. An important property is its recursive relation, ?(t+1) = t?(t), which is instrumental in the derivation and validation of expressions for the moments and variance of the gamma distributed random variable.

Transcript

Please give Ace some feedback