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ECE2191 - Probability and AI for engineers - S2 2026

ECE2191 - Probability and AI for engineers - S2 2026

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This question requires you to write your response on an answer sheet provided by the teaching team. Write your student ID on your answer sheet.

Two normalised sensor outputs and

are jointly modelled by the PDF

For a particular measurement, suppose

is observed to have value , where

. Determine

Show all relevant working.

Hand your answer sheet to the teaching team at the end of the mid-semester test.

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To estimate a constant voltage , an engineer collects IID measurements , each with Two estimators are proposed: Which comparison is correct?

Both are unbiased, but has smaller standard error when .

Both have the same sampling distribution because they estimate the same parameter.

is more precise because it does not combine measurement errors.

Only is unbiased.

is biased because the same data are used to estimate and reduce uncertainty.

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Individual packet delays through a network switch are assumed IID. The individual packet-delay distribution is right-skewed, with A network-monitoring system computes the average delay It issues a warning if Using the Central Limit Theorem, and this Z-table, which statement is most accurate?

The warning probability is approximately 0.0548, and this means that approximately 5.48% of individual packet delays exceed 13.2 ms.

The warning probability is approximately 0.9452, because the table gives .

The warning probability is approximately 0.0548, because and the table gives .

The warning probability cannot be approximated using a normal distribution because the individual packet delays are right-skewed.

The warning probability is approximately 0.4207, because the standard deviation of an individual packet delay is 6 ms, giving and from the table.

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A sensor's acceptable population mean bias is specified as A valid 95% confidence interval for is Which conclusion is best supported?

Exactly 95% of sensors manufactured by this process satisfy the tolerance.

At least 95% of individual sensor errors lie within the acceptable bias range.

The 95% confidence interval lies entirely within the acceptable range, supporting the conclusion that the population mean bias satisfies the specified tolerance.

The probability that every future sensor error is acceptable is 0.95.

The true mean bias is the midpoint .

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A sensor produces measurements where is the constant quantity being measured, represents short-term zero-mean random noise, and is a slowly varying calibration offset that is not assumed to average to zero. An engineer assumes that the may be treated as IID and therefore argues that the Law of Large Numbers (LLN) guarantees converges in probability to as . Which assessment is correct?

The argument is correct because averaging removes any disturbance as the number of measurements increases.

The argument is correct because the Central Limit Theorem guarantees that the sample mean converges to , even when calibration drift is present.

The argument is correct provided more than 30 measurements are collected, since the sample mean can then be treated as approximately equal to .

The argument is incorrect because the drift can make the non-identically distributed, and there is no guarantee that the average drift tends to zero.

The argument is incorrect because the LLN cannot be applied to continuous-valued sensor measurements.

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A circuit measures a noise voltage and quantises each sample to one of three levels. Let denote the resulting normalised quantised sample, where can take only the values , and , each with equal probability. A digital monitoring system computes the power-related metric . Which statement is correct?

and are negatively correlated because squaring removes the sign of .

and are independent because .

and are positively correlated because larger values of produce larger values of .

and are independent because contains no information about the sign of .

and are uncorrelated but dependent.

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The CDF of an analogue sensor output is constant between 2.0 V and 2.5 V: What can the engineer conclude?

Every voltage in this interval has the same positive probability.

The output is uniformly distributed between 2.0 V and 2.5 V.

The output must be discrete.

The probability that the output lies strictly between 2.0 V and 2.5 V is zero.

The PDF is constant and positive over this interval.

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Each packet transmission attempt succeeds independently with probability . Let denote the number of attempts up to, and including, the first success. The first five attempts have failed. What is the probability that more than three additional attempts are required before the first success?

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Two zero-mean Gaussian noise sources and have different standard deviations. An engineer observes that . Which conclusion is valid?

Source has the smaller standard deviation and its probability distribution is more concentrated near zero, although .

The probability that is greater than the probability that .

Source must have the larger variance because its peak is higher.

The PDF of must have a larger total area than the PDF of .

Source has the higher PDF value at zero, but this provides no information about which source has the smaller variance.

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An engineer constructs a 95% confidence interval for the population mean and obtains . Which statement is correct?

There is a 95% probability that the true mean lies inside this particular interval.

Approximately 95% of future observations will lie inside the confidence interval.

The sample mean has a 95% probability of lying inside the confidence interval.

If this sampling procedure were repeated many times, approximately 95% of the resulting confidence intervals would contain the true population mean.

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