The CDF probability for a normal distribution

Question

BlackTom AI · Accepted Answer

The question presented is: The CDF probability for a normal distribution. The provided answer option is: 'is the probability that a value is less than a specific value'.

First, recall what the cumulative distribution function (CDF) represents in probability theory. For a random variable X with distribution F, the CDF at a value x is defined as F(x) = P(X ≤ x). It gives the probability that X takes a value less than or equal to x.

Now, examine the wording of the given option. It states 'the probability that a value is less than a specific value'. This describes P(X < x). In continuous distributions like the normal distribution, P(X ≤ x) and P(X < x) are the same number, because the probability of X equaling any exact value x is zero. Therefore, for a normal distribution, the CDF F(x) equals P(X ≤ x) = P(X < x), and the statement is effectively correct in the continuous case.

However, there is a subtle distinction worth noting: in general (for discrete distributions), P(X ≤ x) and P(X < x) are not the same, because there can be a positive probability at the exact value x. The CDF explicitly uses '≤', not '<', by definition. So, while the given option captures the practical meaning for a continuous distribution like the normal, strictly speaking, the definition uses '≤'.

Since there are no alternative options provided to compare against, we cannot assess other choices. If you encounter a statement about the CDF in a normal distribution, use the precise definition: F(x) = P(X ≤ x). For a continuous normal distribution, P(X ≤ x) = P(X < x), so the practical interpretation aligns with the given option.

In summary: the CDF of a normal distribution gives the probability that the variable takes a value up to x (less than or equal to x). In continuous cases this is equivalent to the probability of being strictly less than x, due to the zero probability of hitting x exactly.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 6. Quiz for Class 6

The CDF probability for a normal distribution

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