Let X be a standard normal random variable and define the event A =\{X 0\}. Which is the following is the conditional probability density function (PDF) of X given A?

Question

Let X be a standard normal random variable and define the event A =\{X > 0\}. Which is the following is the conditional probability density function (PDF) of X given A?

BlackTom AI · Accepted Answer

To evaluate the conditional pdf f_{X|A}(x) where X is standard normal and A = {X > 0}, recall the rule f_{X|A}(x) = f_X(x) / P(A) for x in A, and 0 outside A. Here, f_X(x) = (1/√(2π)) e^{-x^2/2} is the standard normal pdf, and P(A) = P(X > 0) = 1/2.

Applying the formula, for x > 0 we have f_{X|A}(x) = f_X(x) / P(A) = [(1/√(2π)) e^{-x^2/2}] / (1/2) = (2/√(2π)) e^{-x^2/2}.

For x ≤ 0, since the event A occurs only when X > 0, the conditional pdf must be 0. Therefore, the complete conditional pdf is f_{X|A}(x) = (2/√(2π)) e^{-x^2/2} for x > 0 and 0 for x ≤ 0.

Option a. states f_{X|A}(x) = (2/√(2π)) e^{-x^2/2} for x > 0. This matches the correct expression on the support x > 0. However, note that the statement omits the specification that f_{X|A}(x) = 0 when x ≤ 0; without that boundary, it is incomplete but effectively conveys the correct nonzero density on the region where A holds.

In summary, the given form is correct on the domain x > 0, and the full correct definition would include f_{X|A}(x) = 0 for x ≤ 0.

MTH2222 - 2225 - S1 2025 Quiz 6 - Due Week 8

Let X be a standard normal random variable and define the event A =\{X > 0\}. Which is the following is the conditional probability density function (PDF) of X given A?

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