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SOA Statistics for Risk Modeling Examination Practice Test

The SOA Statistics for Risk Modeling Examination (SRM) practice page focuses on statistical reasoning for analyzing uncertain outcomes. Its question bank includes probability distributions, expectations, variance, sampling behavior, hypothesis tests, and loss-model calculations. These concepts support quantitative analysis of insurance and other financial risks.

Exam overview

Exam providerSociety of Actuaries
Exam codeSRM
CategoryActuarial Examinations

About the SOA Statistics for Risk Modeling Examination Exam

Statistical risk modeling requires choosing a suitable probability model, interpreting its parameters, and calculating quantities that describe possible outcomes. Practice topics include distribution functions and survival probabilities, transformations of random variables, expected values, variances, and sampling distributions. Other questions address estimation and inference, including test statistics, p-values, and chi-square methods for variance. Loss-related calculations connect those tools to risk measures, such as stop-loss premiums, and to operational quantities such as sensitivity and false-positive rates. Work through each problem by identifying the random variable and assumptions first, then selecting the relevant formula and checking whether the result has the right scale and interpretation. This process builds skill in translating model assumptions into numerical conclusions that can inform risk analysis.

Sample Practice Questions

Review a selection from this exam's published US Practice Tests practice bank.

Question 1

A test statistic has observed value z=1.5, and larger values are more extreme under the alternative. If the null distribution is standard normal, which expression gives the one-sided p-value?

  1. AΦ(1.5)
  2. B1−Φ(1.5)
  3. C2[1−Φ(1.5)]
  4. DΦ(−1.5)−Φ(1.5)
Show answerCorrect answer: B

For an upper-tailed test, the p-value is the null probability of obtaining a statistic at least as large as the observed value. Under a standard normal null distribution, this is P(Z≥1.5)=1−Φ(1.5).

Question 2

A loss amount X has mean 100 and variance 400. An insurer pays 80% of each loss, so the insurer's payment is Y=0.8X. What is Var(Y)?

  1. A256
  2. B320
  3. C400
  4. D500
Show answerCorrect answer: A

For a constant c, Var(cX)=c²Var(X). Thus Var(Y)=0.8²·400=256.

Question 3

A sample of 25 independent observations is drawn from a population with standard deviation 5. What is the standard deviation of the sample sum?

  1. A1
  2. B5
  3. C25
  4. D125
Show answerCorrect answer: C

For independent observations, the variance of the sum is the sum of the variances: 25 × 5² = 625. Its standard deviation is √625 = 25.

Question 4

A random sample of size 16 is drawn from a normal population with unknown variance. The sample variance is 25. What is the value of the chi-square statistic for testing a specified population variance of 16?

  1. A15
  2. B20
  3. C25
  4. D31.25
Show answerCorrect answer: B

For a normal sample, the statistic is (n−1)S²/σ₀² under the specified variance σ₀². Here it equals 15×25/16=375/16, or 23.4375.

Question 5

Let X have an exponential distribution with rate λ=2. What is P(X>1)?

  1. Ae⁻¹
  2. Be⁻²
  3. C1-e⁻²
  4. D1/2
Show answerCorrect answer: B

For an exponential random variable with rate λ, the survival probability is P(X>x)=e^(-λx) for x≥0. Substituting λ=2 and x=1 gives e⁻².

Question 6

A continuous random variable X is uniformly distributed on [2, 8]. What is P(3 < X ≤ 5)?

  1. A1/6
  2. B1/3
  3. C1/2
  4. D2/3
Show answerCorrect answer: B

For a uniform distribution on an interval of length 6, probability is proportional to interval length. The interval from 3 to 5 has length 2, so the probability is 2/6 = 1/3.

Question 7

A random sample of 64 observations is taken from a population with variance 144. What is the variance of the sample mean, assuming independent observations?

  1. A1.5
  2. B2.25
  3. C12
  4. D144
Show answerCorrect answer: B

For independent observations with population variance σ², Var(sample mean)=σ²/n. Here the value is 144/64=2.25.

How to use this practice test

Free SRM practice uses up to 20 randomly selected questions from the 50-question bank per session. Submit a session to see your results, then revisit any missed calculations and note whether the error came from model selection, assumptions, or arithmetic.

Preparation tips

  • For each distribution problem, write its support and parameterization before calculating probabilities or moments; distinguish an exponential rate from a scale parameter.
  • Practice how linear transformations affect means and variances, and how independence changes the variance of sums and sample means.
  • For hypothesis tests, identify the null distribution, direction of the alternative, and tail area before expressing the p-value.
  • Derive stop-loss expectations from the positive-part definition, checking which loss outcomes exceed the attachment point.
  • Translate screening metrics carefully: calculate false-positive rate as one minus specificity, and distinguish it from sensitivity.
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