Question

1) a). Prove or Disprove: "Every random variable has cumulative distribution". Justify
b). If statement in problem (a) is true, does the existence of cumulative distributions function of a random variable imply the existence of probability density function? Justify.

Answer 1

Since, Every Random Variable we select from some Population. so, there is something chance of selecting that random variable from the whole population

So this chance means the probability of selecting the random variable and also this random variable changes as the values changes and so every random varibale has cumulative distribution

And we know that derivative of Cumulative Distribution function is nothing but probability density function

so, the existence of cumulative distributions function of a random variable imply the existence of probability density function