(Solved): Properties of the CDF The cumulative distribution function, or cdf, of a rand ...

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Properties of the CDF The cumulative distribution function, or cdf, of a random variable $$X$$ is the function $$F(x)=\mathbb{P}(X?x)$$ a. Using the properties of a probability measure, show that $$F$$ is nondecreasing: if $$x?y$$, then $$F(x)?F(y)$$. b. Show that $$F$$ is right-continuous: if $$x_1,x_2,\dots$$ is a decreasing sequence converging to $$y$$, then $$F\left(x_1\right),F\left(x_2\right),\dots$$ converges to $$F(y)$$. c. Show that $$F$$ is normalized: $$\underset{x???}{\lim }F(x)=0$$, and $$\underset{x??}{\lim }F(x)=1$$. Hint: For parts b and c, it may help to revisit question $$1\mathrm{b}$$ of discussion 01.