A question related to Uniform Random Vector

Let U1, U2, . . . ,Um. . . be independent identically distributed random variables that are uni formly distributed on the interval [0,1]. Let N be a random variable that equals the integer value so for which n n+1 H U3 2 6)\" > H U3 1:1 i=1 Where A > 0, and H321 Uz- : 1. Show that N is a Poisson random variable with parameter A. Hint: Apply the transformation: Xi : % log (If. Note that the GDP of Gamma random variable with parameters (k, A) is given by 1 e')' 2:01 (22%