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:

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

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 2,255 words, 15 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 2b

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