Likelihood Function - Exponential Distribution

In summary, X is an exponentially distributed variable with a likelihood function of (t^3)*exp(-3t*(average of X)). The maximum likelihood estimator (MLE) of t can be found by taking the natural log of the likelihood function, taking the derivative, and setting it equal to 0 to solve for t. This will give the value of t that maximizes the likelihood function.
  • #1

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

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  • #2
I've answered your other topic.

1. What is the exponential distribution?

The exponential distribution is a probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.

2. How is the likelihood function used in the exponential distribution?

The likelihood function is used to estimate the parameters of the exponential distribution, such as the rate parameter λ. It is a function of the observed data and the parameters, and it measures the probability of obtaining the observed data given the parameters.

3. How is the exponential distribution related to the Poisson distribution?

The exponential distribution can be derived from the Poisson distribution by considering the time between events rather than the number of events. In fact, if the rate parameter λ is small, the exponential distribution can be approximated by the Poisson distribution.

4. What is the mean and variance of the exponential distribution?

The mean of the exponential distribution is equal to 1/λ, and the variance is equal to 1/λ^2. This means that as the rate parameter λ increases, the distribution becomes more concentrated around the mean and the variance decreases.

5. How is the exponential distribution used in real-world applications?

The exponential distribution is commonly used to model the time between occurrences of events in various fields, such as reliability engineering, queueing theory, and survival analysis. It is also used in financial modeling, where it can describe the time between stock price changes or the time between payments on a loan.

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