- CFA Exams
- CFA Exam: Level I 2021
- Study Session 3. Quantitative Methods (2)
- Reading 9. Common Probability Distributions
- Subject 6. The Normal Distribution

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**CFA Practice Question**

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

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)

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.

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**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. |