Likelihood Function - Exponential Distribution

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SUMMARY

The likelihood function for the given exponential distribution with observations x1=5, x2=3, and x3 > 20 is defined as L(t) = (t^3) * exp(-3t * (avg of X)). The maximum likelihood estimate (MLE) of the parameter t can be determined by taking the natural logarithm of the likelihood function, differentiating it, and setting the derivative equal to zero. This process yields the optimal value of t that maximizes the likelihood function based on the provided observations.

PREREQUISITES
  • Understanding of exponential distribution and its properties
  • Familiarity with likelihood functions and maximum likelihood estimation (MLE)
  • Knowledge of calculus, specifically differentiation
  • Basic statistics concepts, including expectation and cumulative distribution functions
NEXT STEPS
  • Study the derivation of the likelihood function for different distributions
  • Learn about the properties of maximum likelihood estimators
  • Explore the application of the natural logarithm in statistical estimation
  • Investigate the implications of censored data in statistical analysis
USEFUL FOR

Statisticians, data analysts, and students studying probability theory and statistical inference, particularly those focusing on maximum likelihood estimation and exponential distributions.

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Homework Statement


X is exponentially distributed. 3 observations are made by an instrument that reports x1=5, x2=3, but x3 is too large for the instrument to measure and it reports only that x3 > 20 . (The largest value the instrument can measure is 10)

a)What is the likelihood function?
b)What is the mle of t?


Homework Equations


f(x;t)=t*exp(-t*x), E(X)=1/t ,
F(x) = P(X<=x)=1-exp(-t*x)

The Attempt at a Solution


a)(t^3)*exp(-3t*(avg of X))
b) take natural log of a), take derivative, then set equal to 0 and solve for t

Wondering if I am on the right track?
Thanks
 
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