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## Q. 2.2

Let X denote a real number chosen “at random” from the interval [0,10]. Find the cumulative distribution function of X.

## Verified Solution

In this case there is a continuous set of possible values for the random variable, so there are infinitely many values that the random variable X might assume and it is equally likely that X will take on any one of these values. Obviously, this requires that the probability of X being equal to any particular one of the possible values must be zero, because the total probability assigned to the set of all possible values is always unity for any random variable (Axiom 2 of probability theory, also called total probability). Thus, it is not possible to define the probabilities of this random variable by giving the probability of events of the type {X=u}, but there is no difficulty in using the cumulative distribution function. It is given by

 $F_{X}(u)=0$ For $-\infty \lt u \lt 0$ $F_{X}(u)=0.1u$ For $0 \leq u \lt 10$ $F_{X}(u)=u$ For $10 \leq u \lt \infty$

For example, $P(0 \leq X \leq 4) = F_{X}(4) – F_{X}(0) =0.4$.