Question

A grasshopper starts jumping forward on the real axis, starting at 0 in the positive direction. Her jump sizes are independent identically distributed (i.i.d.) random variables, each has exponential distribution with mean 2 units (say, feet). After she jumps over point 9 (for the first time), she starts jumping back, now making i.i.d jumps having exponential distribution with mean 1. 1.1 (15 points) What is the probability that the number of forward jumps it takes her to jump over 9 for the first time (in forward direction), is exactly 4? 1.2 (20 points) What is the probability that the number of backward jumps N it takes her to jump over 9 for the first time (in backward direction), is exactly 4? Hint: The relation between Poisson processes and Exponential distributions may be helpful.

Answer 1