Statistica e mate

This document provides a set of exercises for a Probability and Statistics exam, covering fundamental to advanced topics:

• Elementary Problems (7 points):
  • Combinatorics: Calculating password combinations based on character and digit rules.
  • Binomial Distribution: Determining probabilities of success in independent trials (e.g., hitting a target, dice rolls for even numbers).
  • Hypergeometric Distribution: Probability of sampling satisfactory individuals from a population.
  • Exponential Distribution: Calculating the probability of no failure within a specified time.
• Easy Problems (8 points):
  • Combinatorics (Braille): Counting possible Braille characters, characters with four points, and the probability of specific geometric shapes (square/rectangle) from four points.
  • Independence: Probabilities of hitting a target (once, at most once, at least once) with varying probabilities for each independent shot.
  • Mean and Variance: Relating a binomial distribution to a normal approximation by matching mean and variance.
  • Conditional Probability: Problems involving multiple coins and conditional probabilities related to heads/tails outcomes.
  • Normal Distribution: Calculating probabilities for a normal variable using its CDF (G(z) function).
  • Hypergeometric Distribution (Lamps): Probabilities related to randomly switching lamps in a theatrical setup, and determining the law of total power.
• Regular Problems (9 points):
  • Urn Sampling: Probabilities of drawing specific colored balls from an urn, both with and without replacement, including expected number of draws.
  • Normal and Binomial Distribution: System reliability analysis with components having normal lifetime distributions, focusing on binomial outcomes (number of working components) and associated financial gains/losses.
  • Binary Channel: Probabilities related to data transmission in a non-symmetric binary channel, including error probabilities.
  • Sum of Independent Variables: Calculating mean, variance, and the law of the sum of independent Poisson and Bernoulli variables.
  • Joint Density Function: Problems involving a given joint density function, requiring determination of normalization constant, marginal distributions, independence/correlation, and specific probabilities.
• Difficult Problems (8 points):
  • Bose-Einstein Statistics: Distributing indistinguishable items into distinct boxes, calculating probabilities for specific box counts.
  • Bernoulli Scheme: Analyzing properties of a Bernoulli process, including expected arrival times and conditional probabilities of successes.
  • Gamma Variables: System reliability with components having Gamma-distributed failure times, calculating survival probabilities and density of the system's failure time.
  • Change of Variable and Mixture: Problems involving transformations of exponential mixture distributions (e.g., min(X,Y), X+Y), determining their distributions, joint laws, expected values, and probabilities.