Math Problem Statement

2.5 •••• (With a good understanding of stuff, you can solve this problem without any computation.) Let X1, X2, . . . be iid. random variables each with density 1 2 x 2 e −x for x > 0 and 0 otherwise. Let S0 = 0 and Sn : = X1 + · · · + Xn, and N(t) : = max{n : Sn < t}. a) Determine the density of the random variable S2. b) Find the mass function of the random variable N(t).

Solution

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Random Variables
Convolution of Random Variables
Cumulative Distribution Function (CDF)
Mass Function

Formulas

f_X(x) = (1/2) * x^2 * e^(-x) for x > 0
f_{S_2}(s) = (1/4) * e^(-s) * ∫_0^s x^2 * (s - x)^2 dx
P(N(t) = n) = F_{S_n}(t) - F_{S_{n+1}}(t)

Theorems

Convolution Theorem
CDF for Sum of Random Variables
Properties of Independent Identically Distributed (iid) Random Variables

Suitable Grade Level

Graduate Level