Hi guys, I feel very uncomfortable with PDF, CDF, PMF.

Problem:

I think the plot looks like this:The distribution function of the random variable X is given:

$\displaystyle F(x) = \begin{cases}0 & \text{ if } x < 0\\x/2 & \text{ if } 0\leq x < 1\\2/3 & \text{ if } 1\leq x<2\\11/12 & \text{ if } 2\leq x < 3 \\1 & \text{ if } 3\leq x\end{cases}

$

a) Plot the distribution function.

b) What is P { X > 1/2} ?

c) What is P{2 < X <= 4} ?

d) What is P {X <3 }?

e) What is P{X = 1}?

In this problem we mix both discrete and continuous random variable (R.V.).

For the discrete case, we have PMF and CMF, which they both can give the probability. For continuous, we can only find probability via CMF because P {X = a} (of any particular point) is zero..

We know that $\displaystyle F(x) = P(X \leq x) = \int_{-\infty }^{x} f(t)dt$

and for discrete we have $\displaystyle F(x) = \sum_{x_{i}\leq x} p(x_{i})=\int_{-\infty }^{x}f(t)dt$

For $\displaystyle P(a\leq X \leq b) = \int_{a}^{b}f(x)dx=F(b)-F(a)$

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For (b) What is P { X > 1/2}:

I think the solution is simply 1 - P {x <=1/2} because the probability has to add up to 1.

For (c) P{2 < X <= 4}

Do we just take F(4) - F(2)? Since this is a continuous function, P (x = a) is zero, so is it true that P {2 <= X <= 4} = P {2 < X < 4} = P {2 <= X < 4} ???

For d) What is P {X <3 } ...... similarly, this problem does not include the = sign. Is it equal to P {X <=3 }?? I heard my instructor said something about " P {X < 3- }....

I am not sure how to answer.

For (e) P {X = 1}.... the limit is 2/3 if 1 <= x < 2, so this is a discrete case, right? I'd say P = 2/3, right?

==== I always have difficult time to ask the right questions......but for now I am going to stop here.... thank you for reading and I appreciate for the help! Thanks.