The time that a butterfly lives after emerging from its chrysalis can be modelled by a random variable tt, the model here taking the probability that a butterfly survives for more than tt days as

Answer: The probabilities that either a) the butterfly dies within 7 days or b) the butterfly lives longer than 7 days must add up to one. We know the second probability (lives longer than 7 days) - - it is given. So now set up an equation and solve: P (t > 7) = 36 / 13² = 36/169 - - > 36/169 + x = 1 - - > x = 1 - 36/169 = 133/169 ~ 0.7870 or 78.70% b) 7% is the probability (the expected value) that a butterfly will survive more than t days. Again, this formula is given: 36 / (t + 6) ² = 0.07 - - > (t + 6) ² = 36/.07 - - > t = âš (36/.07) - 6 ~ 16.68 days This is the value of t that will give a probability of 7%. c) This one is a little trickier. Technically the mean lifetime should be calculated as: â"t * p (t) dt where p (t) is the probability distribution function But we have a cumulative probability distribution. I think we need to find the probability distribution from this cumulative one. Which is easy because the cumulative distribution function is an integral of the probability distribution function: P (t) = â"p (x) dx, from x = t, to x = âž Now hopefully it's easy to see that P (t) is an anti-derivative of p (x). Said the other way p (x) is the derivative of P (t) : P' (t) = - 72 / (t + 6) Âł - - > but p (t) shouldn't be negative--also you can convince yourself that it should be positive by using the above value (from t - - > âž ... you get - 0 - a negative = a positive) p (t) = 72 / (t + 6) Âł Now we can calculate the mean lifetime by integrating: 72â"t / (t + 6) Âłdt - - > there's a very easy u-substitution to use here: u = t + 6 - - > du = dt - - > t = u - 6 - - > â" (u - 6) / uÂłdu = â" (1/u² - 6/uÂł) du = - 1/u + 3/u² + C - - > now evaluate from t = 0 to t = âž or from u = 0 + 6 = 6 to u = âž 0 - (-1/6 + 3/36) = 1/6 - 1/12 = 1/12 - - > multiply by the 72 72/12 = 6 days