Question

(1) In week 4, Lecture 1 we saw 7 remarks regarding a continuous random variable, among those pick 4 of them which makes sense for you. (2) State the definition of the cumulative distribution function of a continuous random variable. (see Week 5, Lecture 1) (3) For a continuous random variable, state how one computes P(a < X < b) using both the density function and cumulative distribution function. (see Week 5, Lecture 1) (4) State how we can obtain probability density from cumulative distribution function. (see Week 5, Lecture 1)

no need to answer part a.

Answer 1

Since you have mentioned not to answer first question heres the solution to the other three

2)

Definition The cumulative distribution function Fx of the random variable X


defined on the probability space (S, E, P) is the function Fx: R[0, 1] defined by Fx (2) = P(X<2) = Px(-0, x].



The cumulative distribution function Fx is a nondecreasing, right continuous func-


tion, and satisfies limz+-- Fx(2) = 0 and lim+ Fx(x) = 1.


3)

P(a<=X<=b) using pdf of X, f_x(x) we have,
$P(\leqslant X \leqslant b) = \int_{a}^{b}f_{X}(x)dx$

Now, P(a<=X<=b) using CDF of X, F(x) we have,

$P(a\leqslant X\leqslant b) = F(b) - F(a)$

4)
We can obtain PDF from CDF by differentiating the CDF F(x) of X with respect to x. That is,

$f_{X}(x) = \frac{dF(x)}{dx}$