Stat Sampling

random Variable Party

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Use of random Numbers

🎉 Party Guests: Random Variables

Random variables are like the life of the party in statistics. They are variables that take on different values due to chance or randomness. Think of them as unpredictable characters with their own quirks. In our analogy, each guest at the party represents a different random variable.

🕺 Dancing the Night Away: Variability

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At our random variable party, each guest (random variable) has its own dance moves, or in statistical terms, variability. Some random variables have wild and erratic dance steps, while others are more graceful and predictable. This variability represents the range of values a random variable can take on.

An early example of this occurred in the 1940s when the RAND Corporation generated random numbers using a simulation of a roulette wheel attached to a computer. Calculators use RAND or RND or RAN# to generate random numbers. Random number generators have many applications; for example, statistical sampling, computer simulation and cryptography.

Calculating mean and variance

📈 Probability Function: The DJ's Playlist

Imagine the probability function as the DJ's playlist at our random variable party. It tells us how likely each dance move (value) of a random variable is. For a discrete random variable, this function is called a probability mass function (PMF), and for a continuous random variable, it's called a probability density function (PDF). Let's say we have a random variable X representing the outcome of rolling a fair six-sided die: X can take values {1, 2, 3, 4, 5, 6}. The PMF for X is P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = P(X = 5) = P(X = 6) = 1/6. This means each value on our dance floor (1 through 6) has an equal probability of 1/6, just like each song in the DJ's playlist gets an equal chance to play.

🧮 Expectation (Expected Value): The Most Likely Dance Move

The expectation of a random variable is like trying to predict the most likely dance move at our party. It's calculated as the weighted average of all possible values of the random variable, weighted by their respective probabilities. For our example with the die: E(X) = (1/6) * 1 + (1/6) * 2 + (1/6) * 3 + (1/6) * 4 + (1/6) * 5 + (1/6) * 6 E(X) = (1/6)(1 + 2 + 3 + 4 + 5 + 6) E(X) = (1/6)(21) E(X) = 3.5 So, the expected value of rolling a fair six-sided die is 3.5. This means that, on average, we expect the outcome to be 3.5, even though we can't roll a 3.5 on the die itself.

📏 Variance: Measuring the Spread of Dance Moves

Variance quantifies how spread out the dance moves are on our party floor. It tells us how much the random variable tends to deviate from its expected value. The variance of a random variable X is calculated as: Var(X) = E[(X - E(X))^2] Using our die example: Var(X) = E[(X - 3.5)^2] Var(X) = [(1 - 3.5)^2 * (1/6)] + [(2 - 3.5)^2 * (1/6)] + ... + [(6 - 3.5)^2 * (1/6)] Var(X) = (12.25 * 1/6) + (2.25 * 1/6) + ... + (6.25 * 1/6) Var(X) = (2.0417) + (0.375) + ... + (1.0417) Var(X) ≈ 2.92 So, the variance of rolling a fair six-sided die is approximately 2.92. It tells us that, on average, the dance moves (values) tend to deviate from the expected value (3.5) by about 2.92 units.

💫 Wrapping It Up

remember that Earth is our population, sampling is our spaceship, random samples are our ticket to unbiased insights, and sample means are our dancing random variables.

"A sample mean can be considered a random variable, akin to a dance party with unpredictable moves but governed by certain rules."

A random variable ...

• Earth represents our population, while sampling allows us to study a smaller group to understand the whole.
• Random sampling is crucial to avoid bias and ensure that our sample accurately reflects the population.

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