- #1
bennyzadir
- 18
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Let be $X_1, X_2, ..., X_n, ... $ independent identically distributed random variables with mutual distribution $ \mathbb{P}\{X_i=0\}=1-\mathbb{P}\{X_i=1\}=p $. Let be $ Y:= \sum_{n=1}^{\infty}2^{-n}X_n$.
a) Prove that if $p=\frac{1}{2}$ then Y is uniformly distributed on interval [0,1].
b) Show that if $p \neq \frac{1}{2}$ then the distribution function of random variable Y is continuous but not absolutely continuous and it is singular (i.e. singular with respect to the Lebesque measure, i.e with respect to the uniform distribution).
I would really appreciate if you could help me!
Thank you in advance!
a) Prove that if $p=\frac{1}{2}$ then Y is uniformly distributed on interval [0,1].
b) Show that if $p \neq \frac{1}{2}$ then the distribution function of random variable Y is continuous but not absolutely continuous and it is singular (i.e. singular with respect to the Lebesque measure, i.e with respect to the uniform distribution).
I would really appreciate if you could help me!
Thank you in advance!