continuous distribution

CFA Level I
#1

Consider a random variable X that follows a continuous uniform distribution: 7 ≤ X ≤ 20. Which of the following statements is least accurate?

A) F(12 ≤ X ≤ 16) = 0.307. B) F(21) = 0.00. C) F(10) = 0.23.

Your answer: C was incorrect. The correct answer was B) F(21) = 0.00.

F(21) = 1.00 The probability density function for a continuous uniform distribution is calculated as follows: F(X) = (X – a) / (b – a), where a and b are the lower and upper endpoints, respectively. (If the given X is greater than the upper limit, the probability is 1.0.) Shortcut: If you know the properties of this function, you do not need to do any calculations to check the other choices.

The other choices are true.

  • F(10) = (10 – 7) / (20 – 7) = 3 / 13 = 0.23
  • F(12 ≤ X ≤ 16) = F(16) – F(12) = [(16 – 7) / (20 – 7)] − [(12 – 7) / (20 – 7)] = 0.692 − 0.385 = 0.307

So I got this question in the Schweser self made quiz. Shouldn’t B be a correct answer since page 254 in the first book says a value outside of the parameters should equal 0 and not 1 like this question is telling me?

#2

Answer choice A makes no sense. It’s like writing F(x) = x², so what’s F(12 ≤ X ≤ 16)? (Presumably it’s (12 ≤ X ≤ 16)², which is silly.)

If the density function is f(X), then the (cumulative) distribution is F(X); F(12 ≤ X ≤ 16) is a meaningless expression. The author evidently meant P(12 ≤ X ≤ 16) (which equals F(16) – F(12)).

#3

You are confusing density function (f) and cumulative distribution function (F).

Think tossing of a dice with 6 sides: 1,2,3,4,5,6. What’s the probability that the outcome is LESS than 7? (F = distribution function)

What’s the probability that the outcome IS 7? (f = density function (mass function to be mathematically correct here))

#4

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Here the question is asking for the least accurate, Which is B because if you have given continuous distribution 7 ≤ X ≤ 20 and F(X) for more than 20 it should be =1.

#5

Or maybe F(21) is undefined

#6

It isn’t.

(Somebody who calls himself “MathMan” should know that.)

#7

Prove it; somebody that tries to correct somebody named MathMan should provide evidence. And, somebody trying to prove something to somebody named MathMan that contridicts something MathMan knows to be a true statement should be prepared for disappointment. Did you follow all of that?

#8

LOL!

#9

Thats what I thought.

Actually, my comment about F(21) being undefined is a joke. The domain of the uniform distribution can be interpreted as (-inf,+inf) or as [a,b] (or as many other things). It is largely irrelevent since if the domain is all of R, the density is zero outside of [a,b]. F(21) isn’t even an interresting question, (haha right?).

If c > b and the domain is [a,b] then F© is undefined.

If c > b and the domain is R, then F© = 1.

In either case, F© = 0 is the correct answer because it is clearly wrong. Schweser is very loose in its notation and accuracy. When I took level 1 they had an example where they tried to assert the null rather than reject it. I suspect that they have mostly amature quantitative types writing their review notes and prep exams. This is obvious in their clearly poor notation, e.g. F(a ≤ X ≤ b). The CFA Institute’s quant material isn’t much better. It gets the job done but it is very sloppy and contributes to the confusion and complication of the material.