📊Probability and Statistics Unit 3 – Random Variables & Probability Distributions

Key Concepts

• Random variables assign numerical values to outcomes of a random experiment
• Two main types of random variables: discrete (countable outcomes) and continuous (uncountable outcomes)
• Probability distributions describe the likelihood of different outcomes for a random variable
  • Discrete probability distributions (probability mass function) assign probabilities to specific values
  • Continuous probability distributions (probability density function) describe probabilities over a range of values
• Expected value (mean) represents the average outcome of a random variable over many trials
• Variance and standard deviation measure the spread or dispersion of a random variable's outcomes
• Moment-generating functions uniquely characterize probability distributions and simplify calculations
• Central Limit Theorem states that the sum or average of many independent random variables approaches a normal distribution

Types of Random Variables

• Discrete random variables have countable outcomes (integers, whole numbers)
  • Examples: number of heads in 10 coin flips, number of defective items in a batch
• Continuous random variables have uncountable outcomes within a range (real numbers)
  • Examples: height of students in a class, time until a light bulb fails
• Mixed random variables have both discrete and continuous components
• Bernoulli random variables have only two possible outcomes (success or failure)
  • Used to model binary events (yes/no, true/false)
• Binomial random variables count the number of successes in a fixed number of independent Bernoulli trials
• Poisson random variables model the number of events occurring in a fixed interval of time or space

Probability Distributions

• Probability mass function (PMF) for discrete random variables
  • Assigns probabilities to specific values
  • $P(X = x)$ denotes the probability that the random variable $X$ takes on the value $x$
• Probability density function (PDF) for continuous random variables
  • Describes probabilities over a range of values
  • $f_X(x)$ denotes the PDF for the random variable $X$
• Cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a specific value
  • $F_X(x) = P(X \leq x)$
• Common discrete distributions: Bernoulli, binomial, Poisson, geometric, hypergeometric
• Common continuous distributions: uniform, normal (Gaussian), exponential, gamma, beta

Measures of Central Tendency

• Expected value (mean) is the average outcome of a random variable over many trials
  • For discrete random variables: $E(X) = \sum_{x} x \cdot P(X = x)$
  • For continuous random variables: $E(X) = \int_{-\infty}^{\infty} x \cdot f_X(x) dx$
• Median is the middle value that separates the upper and lower halves of a distribution
  • Less sensitive to outliers than the mean
• Mode is the most frequently occurring value in a distribution
  • Useful for identifying peaks or clusters in the data
• Weighted mean accounts for the importance or frequency of each value
  • $\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}$, where $w_i$ is the weight for the $i$-th value $x_i$

Measures of Variability

• Variance measures the average squared deviation from the mean
  • For discrete random variables: $Var(X) = E[(X - E(X))^2] = \sum_{x} (x - E(X))^2 \cdot P(X = x)$
  • For continuous random variables: $Var(X) = \int_{-\infty}^{\infty} (x - E(X))^2 \cdot f_X(x) dx$
• Standard deviation is the square root of the variance
  • Measures the spread of the distribution in the same units as the random variable
• Coefficient of variation (CV) is the ratio of the standard deviation to the mean
  • Useful for comparing the relative variability of distributions with different means
• Range is the difference between the maximum and minimum values in a distribution
  • Sensitive to outliers and does not provide information about the spread of the data

Properties and Theorems

• Linearity of expectation: $E(aX + bY) = aE(X) + bE(Y)$ for constants $a$ and $b$
• Variance properties:
  • $Var(aX) = a^2Var(X)$ for constant $a$
  • $Var(X + b) = Var(X)$ for constant $b$
  • $Var(X + Y) = Var(X) + Var(Y)$ for independent random variables $X$ and $Y$
• Chebyshev's inequality bounds the probability of a random variable deviating from its mean
  • $P(|X - E(X)| \geq k\sigma) \leq \frac{1}{k^2}$ for $k > 0$, where $\sigma$ is the standard deviation
• Law of Large Numbers states that the sample mean converges to the population mean as the sample size increases
• Central Limit Theorem: the sum or average of many independent random variables approaches a normal distribution
  • Applies regardless of the underlying distribution of the individual random variables

Applications in Real-World Scenarios

• Quality control: model the number of defective items in a production batch using a binomial distribution
• Finance: use normal distribution to model stock price returns and calculate probabilities of price movements
• Insurance: model the number of claims filed within a specific time period using a Poisson distribution
  • Helps determine appropriate premiums and reserves
• Biology: use the exponential distribution to model the time between cell divisions or the survival time of a cell
• Telecommunications: model the number of phone calls arriving at a call center within a given time interval using a Poisson distribution
  • Helps optimize staffing levels and minimize wait times
• Marketing: use the normal distribution to model customer preferences and target products to specific segments
• Reliability engineering: model the time until failure of a component or system using the exponential or Weibull distribution

Practice Problems and Examples

1. A fair six-sided die is rolled three times. Let $X$ be the random variable representing the sum of the three rolls. Find the probability mass function of $X$.
2. The time (in minutes) it takes for a customer to be served at a bank follows an exponential distribution with a mean of 5 minutes. What is the probability that a customer will be served within 3 minutes?
3. The weights of apples in a grocery store are normally distributed with a mean of 150 grams and a standard deviation of 20 grams. What is the probability that a randomly selected apple weighs between 130 and 180 grams?
4. A machine produces bolts with a length that follows a normal distribution with a mean of 10 cm and a standard deviation of 0.5 cm. If a bolt is considered defective when its length is outside the range of 9.5 cm to 10.5 cm, what proportion of bolts produced by the machine are defective?
5. The number of customers arriving at a store per hour follows a Poisson distribution with a mean of 30. Find the probability that more than 35 customers arrive in a given hour.