It is given that P(X = 2) = 0.25 for an exponentially distributed random variable X with rate lambda, where E[X] = 1/la...

Question

It is given that P(X >= 2) = 0.25 for an exponentially distributed random variable X with rate lambda, where E[X] = 1/lambda denotes the expectation of X. What is the value of lambda? (ln denotes natural logarithm)

A. ln2
B. ln(1/4)
C. ln3
D. ln(0.25)

Accepted Answer

lambda = ln2. X ~ Exponential(lambda) so P(X >= t) = e^{-lambda*t}. Setting t = 2 and the probability = 0.25, we get e^{-2*lambda} = 0.25 = 1/4. Taking natural log of both sides gives -2*lambda = ln(1/4) = -ln4 = -2*ln2, so lambda = ln2.

It is given that P(X >= 2) = 0.25 for an exponentially distributed random variable X with rate lambda, where E[X] = 1/lambda denotes the expectation of X. What is the value of lambda? (ln denotes natural logarithm) A. ln2 B. ln(1/4) C. ln3 D. ln(0.25)