The time, in seconds, that a butterfly sits on a leaf of a tree is a random variable X having distribution function

(i) Verify that F is a proper distribution function of a random variable.
(ii) Calculate the probabilities P(X ≥ 2), P(1 ≤ X ≤ 2), P(X ≤ 2|X ≥ 1).
(iii) Calculate the expected value for a butterfly’s visit on a leaf.
(iv) Find the variance of X

(a) by finding the density function of X;

(b) directly from the distribution function F(t) given, by using the result of the previous exercise and check that the two results coincide.

Transcribed Image Text:

F(t) = 1-2-2 F)=1 120. 3