Random Variable: that is neither continuous nor discrete

[nichas]

In the readings for Statistics from Schaum: F(x) is a cumulative distribution function defined as
F(x) = 0 if x < 1
= x/2 if 1<= x < 2
= 1 if x>=2.

It is said that the Random variable for the above distribution function F(x) is neither discrete nor continuous. Can anybody explain this?

 

[David Harper CFA FRM]

Hi nichas,

It's almost continuous (actually it is "right continuous" but not "left continuous") but violates continuous at F(1). At F(1), there is a jump from 0 (imagine the x-axis itself) up to y = F(x) = 0.5. It is that "discontinuous" jump from y=0 to y=0.5 at x=1, that violates continuity (specifically, there is not a limit converging). To the right, it's continuos (and uniform, with a flat pdf and a 45 degree CDF). So, these mixes are common, but in this case, i don't *think* it violates anything special; i.e., it is still a probability. Hope that helps!

 

[nichas]

Hi David,
If I am not wrong then in the above reply you are talking about Continuity w.r.t F(x) the distribution function. But what the author claims is that the Random variable for this F(x) function is neither continuous nor discrete. I want to understand this claim about the Random variable. And if your explanation above is w.r.t the random variable then I failed to understand it.
Kindly help! thanks

 

[David Harper CFA FRM]

Hi nichas,

The distinction escapes me (but I won't claim 100% certainty in regard to author's claim): if f(X) is a continuous probability function, then we call the variable X a continuous random variable (to my knowledge, this follows necessarily). if f(x) is a discrete prob function, we call the random variable X discrete...the distinction escape me because the distribution is a characterization of the random variable...David

 

[nichas]

Hi David,
Thanks for the quick reply,
Agree with your reply but
I am a bit confused with discrete and continuous Random variables(RV).
Is probability function f(x) (curve) being continuous leads to a continuous RV and discrete(means discontinuous) probability function leads to discrete RV.
As per the definition of a Random variable, discrete means RV takes countably finite values then its discrete and if takes uncountably infinite values then that RV is continuous.
How both these things relate if they are the definition of RV.

I appreciate your help!
Thanks Sachin.

 

[David Harper CFA FRM]

Sachin (your user is nichas but not nihcas?),

hmmm...I may be missing your point...sorry...

The RV and the distribution are, in some sense the same thing. It is important to understand this statement: the distribution characterizes the random variable. The density function, for any practical purpose, is just a picture of the random variable.

If it a roll of a 6-sided die, the RV is discrete; it density (pdf) is a discrete plot of six points.
If we refer instead to to throw of a shot-put and the measurement of its distance, the random variable (where does it land?) is continuous; the ensuing pdf is continuous.

On your image - the random variable is on the X-axis. It "arrived" to us as non-continuous, because the blue line jumps; or if you like, we can look at the pdf/CFD (f(x) or F(X)) and infer about the random variable "it is not continuous." But beyond that, i am not sure the semantics are helpful; i mean, this is a right continuous f(x), it's "mostly continuous" ....

or another way to look at:

in regard to the x-axis, the x values are the states of the random variable
(e.g., six sided die: 1,2,3,4,5,6; shot-put distance X meters to Y meters). Already in the x-axis, we have started to decide whether it is continuous or random.

The y-axis is just our estimate of the probability that each state will occur. The y-axis "puts shape" to the nature of the random variable.

Perhaps that helps, to dwell on the nature of the X-axis (all possible states of the random variable) and the Y-axis (the corresponding probability for each state, in the case of the pdf)?

 

[nichas]

Hi David,
Thanks that helped!
Thankyou very much!

Good observation David,'nichas' is the reflection about the central letters 'ch' and hence is nichas and not nihcas since those transformations are already applied by people I wanted a different logic.

Thanks again.
Sachin.