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Chapter 6 Continuous random variables • probability density function (f (x)) the probability distribution function of a continuous random variable (analogous to the probability mass function for a discrete random variable), which takes only positive values at any possible value of the variable, and such that the total area under its graph equals 1; P (a ≤ X ≤ b) = the area under the graph of f (x) over the interval a ≤ X ≤ b • cumulative distribution function (F (x)) the function equal to the area under the graph of f (x) over the interval X ≤ x; it follows that P (a ≤ X ≤ b) = F (b) − F (a) 1 Chapter 6 Normal random variables • normal distribution – by far the most common probability distribution in statistics, and central to the underlying theory – characterized by the iconic bell-shaped curve – completely determined by knowledge of two parameters, its mean µ and standard deviation σ – symmetric about its mean value – the region lying within one standard deviation of the mean is the interval where the bell curve is concave down – a normal variable X can take any real number value (−∞ ≤ X ≤ ∞), but X is less likely to take values further away from the mean, i.e., the distribution is asymptotic (meaning that in either direction the tails of the distribution curve approach – without ever touching – the horizontal axis) 2 Chapter 6 • normal density function Where X has mean value µ and standard deviation σ, 1 −(x−µ)2 /2σ 2 f (x) = √ e σ 2π • uniform distribution parameters if X is a normal random variable, then E(X) = µ V ar(X) = σ 2 SD(X) = σ The percentages listed in the Empirical Rule come from computing the normal probabilities: P (µ − σ ≤ X ≤ µ + σ) = 0.6826 P (µ − 2σ ≤ X ≤ µ + 2σ) = 0.9544 P (µ − 3σ ≤ X ≤ µ + 3σ) = 0.9972 3 Chapter 6 The standard normal random variable Any normal random variable X with mean µ and standard deviation σ can be transformed into the standard normal random variable Z by means of the rescaling formula X −µ . Z= σ Whenever X takes a value x, Z will take the corresponding standardized value x−µ z= . σ By construction, Z always has mean value 0 and standard deviation 1. 4 Chapter 6 Exponential random variables • exponential random variable measures the time X between Successes in a Poisson process, a process in which events occur continuously and independently at a fixed rate of λ Successes per unit time (λ is called the rate parameter) • exponential probability density function since we must have X ≥ 0, the density function is undefined for negative values; for x ≥ 0, f (x) = λe−λx • exponential cumulative density function where X ≥ 0 is an exponential random variable, F (x) = P (X ≤ x) = 1 − e−λx • exponential distribution parameters if X is an exponential random variable, then 1 1 and SD(X) = σ = E(X) = µ = λ λ 5 Chapter 6 Lognormal random variables • lognormal random variable a random variable Y whose logarithm X = ln Y is normally distributed; useful for describing some positive variable quantities with positive skew: incomes, prices, times between Successes in situations where the rate of failure is not constant over time, etc. • lognormal probability density function where X is a normal random variable with mean µ and standard deviation σ and Y = eX , then Y is lognormal, and for any y > 0, 1 2 2 f (y) = √ e−(ln y−µ) /2σ yσ 2π • lognormal parameters where X is a normal random variable with mean µ and standard deviation σ and Y = eX , then Y is lognormal, with µ+ 21 σ 2 E(X) = µY = e q SD(X) = σY = (eσ2 − 1)e2µ+σ2 6