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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 continous but not absolutely continous 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!