Should we use discrete or continuous variable formula to solve the below question?

Question

I have used the formula of Discrete variable. It has specified in the question "...over a period of time." so I am confused if it should be continuous variable formula?

Answer 1

Addressing only the parts about the CDF: I suggest you make a table with three columns headed $x$, PDF or $f_X(x),$ and CDF or $F_X(x),$ and rows for $x = 5, 6, 7, 8, 9.$ Then a plot of the CDF against x, will look somewhat like the following.

The heavy dots are from your table. Strictly speaking, only the horzontal line segments are part of the CDF; the vertical line segments are often added (sometimes as dotted lines), to make the graph easier to follow.

Trying to clear up possible points of confusion:

• While the CDF of a random variable is defined for all values of $x$ on the real line, we usually plot the CDF in an interval of $x$'s where the random variable $X$ has positive probability.

• The values taken by the CDF are always in the interval $[0,1].$ For your random variable $X,$ we have $F_X(-3) = F_X(4) = F_X(4.999) = 0,\, F_X(5) = F_X(5.5) = 0.2,$ and $F_X(9) = F_X(27) = 1.$

• This is the CDF of a discrete random variable. All increases as one moves from left to right are 'jumps'; the CDF is a 'pure jump function'.

• The CDF of a continuous random variable has no jumps at all.

In part (b),

$$P(X < 8) = P(X \le 7) =F_X(7)= P(X = 5) + P(X = 6) + P(X = 7),$$ where $F_X$ is the CDF of $X.$ Also,

$$P(6 < X < 8) = P(X = 7) = F_X(7) - F_X(6).$$

This is the height of the vertical line segment on the graph above $X=7.$