1. Show that the sequence (Xr)new of independent random variables converges in distri-
(a) the probability density function of Xn is
fn(x) = 0 72 : otherwise;
(b) the probability mass function of X, is
= = =
P(IX =}) 0
2. The sequence of i.i.d. random variables with
forms the scaled random walk
3 X, n 8 2
Show that there is a random variable W such that
What is its law?
3. A random variable X is called symmetricif x and - X are identically distributed. Show
that X is symmetric if and only if the imaginary part of its characteristic function is
identically zero. Hint: Consider ((xx-0x). -
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