Well, it looks like that you are right - that there should be \lambda_i in the numerator rather than \lambda_1. One suggestion that might help: if these z_i's are independent exponential random variables with parameter(s) lambda_i's then check the context for any relations between these lambda's parameters.

thank you for your suggestion. Yes, it has been assumed that z_i's are independent random variables with means (1/lambda_i). can you explain how does the result simplify for this assumption? Steftcho P. Dokov

If there isn't any additional information from "some context for any relations between these lambdas" then \lambda_1 is most likely a typo, thus the correct one expression should be \lambda_i in numerator.