Original practice · fully worked
Original variant: normal output within a conditioned range
Daily output X is normal with mean 50 and standard deviation 10. Let Phi denote the standard normal CDF. Determine P(X≤40 | X>20).
- A
1-Φ(1) - B
[Φ(-1)-Φ(-3)]/Φ(3) - C
Φ(-1)/Φ(-3) - D
[Φ(3)-Φ(1)]/Φ(1) - E
Φ(1)-Φ(-3)
Answer and calculation
The standardized interval is -3<Z≤-1 and the conditioning event has probability Φ(3). Their ratio is [Φ(-1)-Φ(-3)]/Φ(3), choice B.
Setup
Setup
Standardizing 20 and 40 gives boundaries negative three and negative one.
X=20↦Z=−3,X=40↦Z=−1 Model
Model
The numerator is the probability between those boundaries, while the denominator is the probability above negative three.
Pr(X≤40∣X>20)=Pr(Z>−3)Pr(−3<Z≤−1) Compute
Compute
Writing both probabilities with the standard normal cumulative distribution gives the expression in choice B, numerically about 0.157518.
Pr=Φ(3)Φ(−1)−Φ(−3)=0.157518 Answer
Answer
The conditional probability is the expression in choice B.
Φ(3)Φ(−1)−Φ(−3)(B) 