CFA Practice Question

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CFA Practice Question

For x, a normal random variable, which of the following is (are) false?

I. P(x < mean) = 50%
II. P(x < 38) = P(x < 38)
III. The higher the standard deviation the more spread out the distribution.
IV. P(35 < x < 50) = P(x <50) - P(x < 35)
Correct Answer: I

P(x < mean) = 50%. A normal distribution is centered (symmetrical about) the mean. The area under the curve to the left of the mean, P(x <= mean), is 0.5. The area under the curve to the right of the mean, P(x >= mean), is 0.5. The total area under the bell-shaped curve is 1.

User Contributed Comments 18

User Comment
tony1973 I believe the answer tries to emphasize that x can be equal to the mean, therefore P(x < mean) is actually less than 50%. You can have a normal distribution in which some observations' values are exactly the mean, therefore P(x<mean) must be less than 50% if it is a normal distribution.
mordja If P(x<= mean) = P(x<mean) + p(x=mean)
If we learnt in the previous question that the probability of any actual value equals zero, then p(x=mean)=0
Therefore it follows that P(x<=mean) = p(x<mean)
Thus p(x<mean) must equal p(x>mean),
therefore p(x<mean) + p(x>mean) = 1 (as the area under the distribution is 1),
thus p(x<mean) = 1/2.......
jmcohen87 i agree
DannyZhou Am I insane or are you guying not reading the question. (I) is correct. But the question is asking "which of the following is/are false?". So the correct answer is none of them.
Poorvi DannyZhou - the answer is that I is false. This is because P(x<=mean) is 0.5 and not P(x<mean) is 0.5
kenchew1 I is false because a normal distribution can be skewed and have kurotisis. If it is skewed, P(x<median) = 50%, not P(x<mean).
LionHero I is true since for any single point on a continuous distribution, the probability is 0; thus P(X<mean) = P(X<=mean)
gazza77 Kenchew1, a normal distribution cannot be skewed and has no excess kutosis
gazza77 is a nornal distribution also a continuous distribution?

If so, say for example p(x=4)=0
therefore p(x=mean)=0

therefore P(x<mean) = P(x<=mean)

therefore I is true and all the choices are false
jmumm I, too, agree with the mordja's logic.
tll936 is that mean the ans. false?
eddington Analyst Notes people, help on this one......
CFAIcand15 any further insight on this one?
nfressell2 It may be rounded up to 50% but since it cannot contain the mean it is less than 50%. I.e it is 49.99999999999999999%
Fabulous1 MordJa, your logic is funny. If you'd follow it properly then since all the single values are zero, also the sum of those values would be zero.
Even if a single value is zero it still has to be included in the calc of the appropriate integral to solve for the prob of an interval.
Akiva Fabulous, if the single value is zero, it does not mean that the sum of them will be zero.
Mordja, jmcohen87, LionHero, Gazza77, +1.
I is true.
myron Akiva: if the single value is zero, it does not mean that the sum of them will be zero.

What did you really want to say?
EMMADADA I agreed with DannyZhou. All of them are correct. So which of the following is(are) false? The correct answer should be, none of them.
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