TASK LIST NO. 5: Limit Theorems and Approximations

Task 1

Poisson Theorem – rare events The defect rate of a certain electronic component is \(p=0.002\). A batch of goods contains \(n=1000\) such components. Calculate the probability that in this batch: 1. there will not be a single defective item, 2. there will be at most 3 defective items.

Hint: Since \(n\) is large (\(n \ge 100\)), \(p\) is small (\(p \le 0.1\)), and \(np \le 10\), the Poisson approximation with parameter \(\lambda = np\) should be used.

Task 2

De Moivre-Laplace Theorem (Integral) – coin tossing We toss a symmetric (fair) coin \(n=100\) times. Calculate the probability that the number of obtained heads will fall within the interval \(\langle 45, 55 \rangle\).

Goal: Application of the normal approximation to the binomial distribution for a large number of trials.

Task 3

Sample size selection (Inverse CLT) The probability of a boy being born is \(p=0.515\). How many times must the experiment (birth) be repeated to assert with a probability of at least \(0.95\) that the frequency of boys in the sample differs from the probability \(p\) by less than \(0.01\)?

Hint: Use the inequality \(P(|\frac{k}{n} - p| < \varepsilon) \ge 1 - \alpha\) using the normal cumulative distribution function.

Task 4

Central Limit Theorem – sum of rounding errors We add together \(n=100\) numbers, each of which has been rounded to the nearest integer. The rounding errors are independent random variables uniformly distributed on the interval \((-0.5; 0.5)\). Calculate the probability that the error of the sum of these numbers (in absolute value) does not exceed \(3\).

Comment: This is a classic example of summing variables with a non-normal distribution (here: uniform), which result in a normal distribution.

Task 5

Central Limit Theorem – elevator load A freight elevator can carry a load of up to \(2000\) kg. \(25\) people enter the elevator. The weight of a random passenger is a random variable with an expected value of \(75\) kg and a standard deviation of \(10\) kg. Calculate the probability that the total weight of the passengers does not exceed the elevator's permissible load.

Task 6

Operating time approximation – exponential distribution A laptop battery has an operating time that is a random variable with an exponential distribution with a mean of \(4\) hours. A user plans a long fieldwork session and takes \(36\) charged batteries with them (replacing them immediately after depletion). What is the probability that the total operating time on this set of batteries will exceed \(150\) hours?

Task 7

Local De Moivre-Laplace Theorem Event \(A\) occurs in a single experiment with a probability \(p=0.6\). We repeat the experiment \(n=600\) times. Calculate the probability that event \(A\) occurs exactly \(370\) times.

Hint: For large \(n\), the point probability \(P(X=k)\) is approximated by the value of the normal distribution density function.

Task 8

Comparison of approximations The probability of success in a single trial is \(p=0.1\). We perform \(n=30\) trials. Calculate the probability of obtaining exactly 2 successes using: 1. the exact Bernoulli formula, 2. the Poisson approximation, 3. the local De Moivre-Laplace theorem. Compare the results.

Task 9

Chebyshev's Inequality A random variable \(X\) has an expected value \(E(X)=10\) and a variance \(D^2(X)=4\). Without knowing the exact distribution of this variable, estimate (provide a lower bound for) the probability that the variable \(X\) takes a value from the interval \((4, 16)\).

Task 10

Statistical application – sample mean A random sample of size \(n=100\) was taken from a population in which feature \(X\) has a distribution (not necessarily normal) with a mean \(\mu=100\) and a variance \(\sigma^2=25\). Calculate the probability that the arithmetic mean of this sample \(\bar{X}\) will be less than \(99\).