# Random Variable Y following uniform distribution with parameter Random X that follows geometric.

Random variable X follows geometrical distribution with p=1/4. Random variable Y follows uniform distribution in [-X,X]. I'm looking for P(Y>3/2) and also P(X=2|Y>3/2).I know for a fact that Σ(from k=1 to infinity)zk/k=-log(1-z) for |z|<1.

The probability that $$Y \gt 3/2$$ is actually the probability that $$3/2 \lt Y \lt X$$.
If $$X \lt 1$$ then it is impossible for $$Y$$ to be greater than $$3/2$$.
Therefore $$X$$ must be at least $$2$$ for the probability to even make sense. So we have $$\mathbb{P} \left(Y \gt \frac32 \right) = \sum_{x=2}^\infty \left( \frac14 \right)\left( \frac34 \right) ^{x-1} \left( \frac{x - 3/2}{2x}\right)$$
• No, it is possible for $Y>1.5$ when $X=2$ – Graham Kemp May 20 at 23:19
• @Graham Kemp True, I simply misread it as $Y > 3$ when I solved the problem. Will fix – WaveX May 20 at 23:20
• This sum converges to about $.2159$ – WaveX May 22 at 17:39