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TASK LIST NO. 4: Selected Random Variable Distributions

Task 1

The probability of failure of experimental equipment in a single experiment is \(p=0.02\). Experiments can be performed any number of times. Calculate the probability that the second failure:

  1. occurs at the tenth experiment,
  2. does not occur in the first ten experiments.

Task 2

The probability that a product subjected to a test fails the test is \(p=0.01\). Calculate the probability that among 200 such products (independently tested), at most 2 will fail the test.

Hint: Since \(n=200\) is large and \(p=0.01\) is small, use the Poisson approximation with parameter \(\lambda = np\).

Task 3

The time (in minutes) between consecutive subscriber calls at a certain telephone exchange is a random variable with an exponential distribution with parameter (expected value) \(\lambda=2\). Calculate the average time between consecutive calls and the probability that a call occurs before 3 minutes elapse.

Task 4

The failure-free operation time \(X\) of a certain device has an exponential distribution with parameter (expected value) \(\lambda=5\). Calculate:

  1. the average failure-free operation time of the device,
  2. the median,
  3. the probability that the failure-free operation time of the device is at least 5 hours.

Task 5

The interval between consecutive graduations of a stopwatch scale is \(0.1\) s. Time on this stopwatch is read with an accuracy of a whole graduation. Assuming a uniform distribution of the time reading error, calculate the probability that the time was measured with an error exceeding \(0.02\) s.

Hint: The density of the uniform distribution is constant in the interval \((-0.05; 0.05)\).

Task 6

An automated machine produces 10-gram weights. The mass measurement errors of these weights have a normal distribution with an expected value \(\mu=0\) g and a standard deviation \(\sigma=0.01\) g. Find the probability that the mass measurement will be performed with an error not exceeding \(0.02\) g.

Task 7

Let the random variable \(X\) have a distribution \(N(\mu, \sigma)\). Calculate the probability \(P(|X-\mu| < k\sigma)\) for:

  1. \(k=1.96\) (confidence level 0.95),
  2. \(k=2.58\) (confidence level 0.99).

Task 8

A certain measuring instrument makes a systematic error of \(1\) m in the direction of overestimating the measurement and a random error with a distribution \(N(0; 0.5)\).

  1. Calculate the average value of the measurement error.
  2. Determine the probability that the error with which the examined objects are measured does not exceed \(2\) m.

Task 9

The strength of steel ropes from mass production is a random variable with a distribution \(N(1000 \text{ kg/cm}^2, 50 \text{ kg/cm}^2)\). Calculate what percentage of ropes has a strength less than \(900 \text{ kg/cm}^2\).

Task 10

Determine and sketch the cumulative distribution function of the Rayleigh distribution, whose density is given by the formula:

\[ f(x) = \begin{cases} \frac{2}{\lambda} x \exp(-\frac{x^2}{\lambda}) & \text{for } x > 0 \\ 0 & \text{for } x \leqslant 0 \end{cases} \]

(where parameter lambda is related to noise variance).

Then calculate the median of this distribution.

Hint: This distribution is often used in telecommunications to model signal fading.