core practice · coding problem
Normal tail probability
Latency is Normal(μ=100ms, σ=20ms). Approximate P(X > 140ms) using the z-score and a standard normal table (Φ(2) ≈ 0.977). What fraction is within 1σ of the mean?
The mechanism
A random variable is a function from outcomes to numbers. Expectation is the long-run average; variance measures spread. The CLT says sums of many independent pieces look Gaussian — which is why normal distributions are everywhere.
Expected evidence
z=(140−100)/20=2; P(X>140)≈1−0.977=0.023. Within 1σ: ≈68%.
Try this drill in the practice workspace →
Hints when you need them
- z = (x−μ)/σ.
- Tail probability is 1−Φ(z) for upper tail.
Watch for these mistakes
- Confusing the PDF value with a probability (for continuous RVs)
- Assuming small samples look like their population distribution
- Using variance when you mean standard deviation in an explanation