Problem regarding the exponential distribution with mean

Reference no: EM13983716

1. Let X, X1, X2, X3,... be random variables that are positive and integer valued. Show that Xn """ X if and only if for every integer k.

lim n→∞ P(Xn = k) = P(X = k)

2. Let Z1, Z2,... be iid random variables with density f . Suppose that P(Zi > 0) = 1 and that λ = limx↓0 f (x) > 0. Let Xn = n min{Z1,..., Zn}. Show that Xn """ Z where Z has an exponential distribution with mean 1/λ.

Reference no: EM13983716

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