Conditional Probabilities for Random Variables

A policyholder filed 2 claims last year. The underwriter needs the posterior probability this person is high-risk, that determines the premium increase. Conditional probability in the random variable setting drives every Bayesian classification on Exam P.

For events defined through $X$, such as $A = \{X > 5\}$ and $B = \{X > 2\}$:

$$P(X > 5 \mid X > 2) = \frac{P(X > 5)}{P(X > 2)}$$

The intersection simplifies: $\{X > 5\} \subset \{X > 2\}$, so $\{X > 5\} \cap \{X > 2\} = \{X > 5\}$.

Given event $B$ with $P(B) > 0$:

$$p_{X|B}(x) = \frac{P(X = x, B)}{P(B)}$$

If $B = \{X > a\}$, this is truncation and renormalization:

$$p_{X|X>a}(x) = \frac{p_X(x)}{P(X > a)}, \quad x > a$$

HIGH-FREQUENCY: The law of total probability and Bayes' theorem applied to random variables are tested constantly, often via a "prior on a parameter."

Common mistakes

• Forgetting to normalize. The posterior is likelihood times prior divided by total probability. Reporting the numerator (0.0251) instead of the posterior (0.4965) is a common error. Trap: 0.0251.
• Wrong Poisson formula. $P(X = k \mid \lambda) = e^{-\lambda}\lambda^k/k!$. Forgetting the $k!$ for $k = 2$ doubles the answer. Trap: 0.5020 instead of 0.2510.
• Applying memorylessness to non-exponential distributions. Given $X > 100$ for a Pareto, you cannot say $P(X > 250 \mid X > 100) = P(X > 150)$. Trap: 0.8133 instead of 0.7037.

Bottom line

• Same formula, variable-defined events: $P(A \mid B) = P(A \cap B)/P(B)$
• Nested events: when $A \subset B$, $P(A \cap B) = P(A)$; never multiply the two survival probabilities
• Memoryless property: $P(X > s+t \mid X > s) = P(X > t)$, exponential and geometric ONLY
• Left-truncation: $f_{X|X>a}(x) = f(x)/S(a)$, rescale density by survival at the truncation point

Exam shortcut

"Given survival to time $t$" or "given loss exceeds $d$", immediately check: is it exponential? If yes, invoke memoryless. If not, compute $S(s+t)/S(s)$. "Bayes = Likelihood times Prior, then Normalize." Write L-P-N at the top. "Memoryless means Exponential or Geometric, nothing else." Pareto, Weibull, uniform, normal: full conditional required.

The full lesson (about 2,608 words, 17 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 2b

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