Let X be a continuous random variable with density, and let
X1, X2 be two independent draws from X. Then,
not usually is it the case that the random variable 2X is
distributed as X1 + X2. However, the Cauchy
density, which is given by the form
, possesses the following property; X1+X2
has the same distribution as the random variable 2X.
a. Let X be a binomial. Argue, based on the properties of the binomial distribution, that X1 + X2 is not equal to 2X.
b. Now, let X be a Cauchy. Based on the property described earlier, argue without any explicit calculation that the variance of the Cauchy distribution is necessarily infinite.
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Let X be a continuous random variable with density, and let
X1, X2 be two independent...
Let X be a continuous random variable with density fx such that X has the same...
Let X be a continuous random variable with density fx
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Question 3: Let X be a continuous random variable with
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Let X1 and X2 be random variables, not necessarily independent. Show that E [X1 + X2] = E [X1] + E [X2]. You may assume that X1 and X2 are discrete with a joint probability mass function for this problem, while the above inequality is true also for continuous random variables.
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Question 3: Let X be a continuous random variable with
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