Suppose X1,X2,….,Xn is a random sample from an exponential distribution with mean λ.If λ1 and λ2 are respectively the maximum likelihood estimators of the mean and the median of the underlying distribution ,then-
c) λ1<λ2 and λ1>λ2 both possible
We have (X1,X2,…..,Xn) IID from Exp(mean= λ). Now λ1 is MLE of the mean that is in case of Exponential setup we have λ1=X bar (Arithmetic mean).
Similarly λ2 is the MLE of the median where median is λln2 that is the MLE of the median is λ2=X bar*ln2.
Since ln2 < 1 we have λ1>λ2 which is option d)