Consider two identically distributed zero - mean random variables $$U$$ and $$V.$$ Let the cumulative distribution functions of $$U$$ and $$2V$$ be $$F(x)$$ and $$G(x)$$ respectively. Then for all values of $$x$$

A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

A fair coin is tossed independently four times. The probability of the event ''The number of times heads show up is more than the number of times tails show up'' is

Consider two independent random variables $$X$$ and $$Y$$ with identical distributions. The variables $$X$$ and $$Y$$ take values $$0, 1$$ and $$2$$ with probability $$1/2,$$ $$1/4$$ and $$1/4$$ respectively. What is the conditional probability $$P(X+Y=2/X-Y=0)?$$