# Probability Models 1. Show that the sequence (Xr)new of independen...

## Transcribed Text

Probability Models 1. Show that the sequence (Xr)new of independent random variables converges in distri- bution where (a) the probability density function of Xn is fn(x) = 0 72 : otherwise; (b) the probability mass function of X, is = = = = P(IX =}) 0 : otherwise; 2. The sequence of i.i.d. random variables with = and = forms the scaled random walk '8n 3 X, n 8 2 i-1 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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