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ECE2191 - Probability and AI for engineers - S2 2026

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A two-axis accelerometer produces measurement errors and , which are collected into the vector Its covariance matrix is What does either off-diagonal entry, 0.015, in the covariance matrix represent?
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A calibration experiment estimates a mean sensor bias using the confidence interval where the confidence level is fixed and the population standard deviation is known. The experiment currently has margin of error . The engineer now requires a margin of error at most . Ignoring integer rounding, what minimum multiplicative change in sample size is required?
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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.

A power-electronics cooling fan is assumed to have an exponentially distributed operating lifetime

, measured in hours, with PDF

The manufacturer's stated mean operating lifetime is 2000 h.

(a) Determine the rate parameter , including its units.

(b) Determine the variance and standard deviation of .

(c) An engineer claims that because the mean lifetime is 2000 h, most fans should fail close to 2000 h. Briefly explain why the exponential model does not support this interpretation.

Show all relevant working. (Note: You do

not need to derive and/or from first principles if you know the corresponding formulae for an exponential random variable.)

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

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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?
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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?
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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?
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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?
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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?
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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?
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