Consider two independent random variables X1 andX2 having the same Cauchy distributionf(x) = 1π(1 + x2)for − q < x < qFind the probability density of Y1 = X1 + X2 by usingTheorem 1 to determine the joint probability density ofX1 and Y1 and then integrating out x1. Also, identify thedistribution of Y1.

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Consider two independent random variables X1 and
X2 having the same Cauchy distribution
f(x) = 1
π(1 + x2)
for − q < x < q
Find the probability density of Y1 = X1 + X2 by using
Theorem 1 to determine the joint probability density of
X1 and Y1 and then integrating out x1. Also, identify the
distribution of Y1.

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