Ap Stat: Random Variables

AP Stat Chapter 7 Random Variables By Andrew Staniforth Discrete and Intro • Random variable – a variable whose value is a numerical outcome of a random phenomenon - Denoted by a Capital letter • A discrete random variable X – countable number of possible values and how probablities are assigned to those variables • Probability distribution of X – lists the values and probabilities • Probabilities are between 0 and 1; and must all add to 1 • Probability Histogram – a pictorial representation of the probability distribution of a discrete variable; Outcome on X axis and Probability on Y axis Continuous • Continuous random variable X – takes all values in an interval of numbers • Probability distribution of X – described by a density curve • Probability of any event is the area under the density curve and above the values of X that make up the event • No histogram – Density Curve instead Formula’s and Notation μ = mean; σ = standard deviation P^~N(μ, σ) Z=(X-μ)/σ=(P^- P)/√(P(1-P)/n) √Variance = Standard Deviation X P X^2

1 .1 1

E(x) 1*.1 E(x^2) (1^2)*. 1

2 .85 4

3 .05 9

2*.85 3*.05 Add these values for the mean(μ) (2^2)*.85 (3^2)*.05 Variance = E(X^2) – μ^2

The Law Of Large Numbers – the average results of many independent observations are stable and predictable • Draw Independent observations at random from a population with mean μ • As “n” increases the statistical mean (from a sample) of the observed values approach μ (from the population)

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How Large is Large? – Depends on variation; more variation the larger “n” has to be!

Rules for Means Add the means Rules for Variance Square it then add! ALWAYS ADD! ALWAYS GETS LARGER!