Question

In: Statistics and Probability

(R code needed) Lifetimes of electronic components manufactured by an electronic company are assumed to follow...

(R code needed) Lifetimes of electronic components manufactured by an electronic company are assumed to follow an Exp(λ) distribution with mean 1/λ. A random sample of 30 lifetimes in years was obtained and shown below: 5.1888 3.6757 4.5091 7.1320 1.3711 1.6454 2.1979 3.8805 0.5290 2.3796 3.2840 3.6678 0.6836 7.9914 12.9922 2.6192 0.3593 5.0234 0.2240 7.5862 0.1172 0.0618 2.6203 16.2319 17.2107 0.8101 8.9368 2.0752 0.9925 1.0187

The data can also be found in life2018.csv.

(a) Find the moment estimate of λ. [4 points]

(b) Find the MLE of λ. Does the MLE match the moment estimate? [4 points]

(c) Let Sλ(2) = exp(−2λ) denote the survival function at time y = 2. Find the MLE of Sλ(2). [4 points]

(d) Find the moment estimate of Sλ(2). [4 points]

Expert Solution

The life times rae exponentially dostributed. .

a) The moment estimate of is found as

The sample mean of observations is

The expected value of the distribution is .

b) The MLE of base on the sample is found as follows. The likelihood function is

The loglikelihood is

Diffeerntiating and equating to 0,

The moment estimator and MLE are the same.

Use R to find the sample mean.

T <- c( 5.1888 ,3.6757, 4.5091, 7.1320 ,1.3711, 1.6454, 2.1979, 3.8805, 0.5290, 2.3796, 3.2840, 3.6678 ,0.6836, 7.9914, 12.9922 ,2.6192 ,0.3593, 5.0234, 0.2240, 7.5862 ,0.1172 ,0.0618, 2.6203, 16.2319 ,17.2107, 0.8101 ,8.9368, 2.0752, 0.9925, 1.0187)
length(T)
mean(T)
1/mean(T)

c) The MLE of is

d) Since MOM estimator and MLE are same,

orchestra answered 1 year ago

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