Continuous random variables

Transcription

1 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

2 Uniform random variables uniform distribution assigns probability uniformly across all values of a continuous random variable X, that is, any interval of values of X of a given size is assigned the same probability: for any numbers r, s, t for which the following equation makes sense, P (s X t) = P (r + s X r + t). uniform density function Where X takes on values in the interval a x b, f(x) = { 1 b a if a x b 0 if x < aor x > b uniform distribution parameters if X is a uniform random variable, then E(X) = µ = a + b V ar(x) = σ (b a) = 1 (b a) SD(X) = σ = 1

3 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) 3

4 normal density function Where X has mean value µ and standard deviation σ, f(x) = 1 σ /σ e (x µ) π normal distribution parameters if X is a normal random variable, then E(X) = µ V ar(x) = σ SD(X) = σ 4

5 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 Z = X µ. σ Whenever X takes a value x, Z will take the corresponding standardized value z = x µ σ. By construction, Z always has mean value 0 and standard deviation 1. The Empirical Rule The percentages listed in the Empirical Rule come from computing the normal probabilities: P (µ σ X µ + σ) = P (µ σ X µ + σ) = P (µ 3σ X µ + 3σ) =

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 E(X) = µ = 1 λ and SD(X) = σ = 1 λ 6

7 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 Y is lognormal, with X = ln Y normal having mean µ and standard deviation σ, then for any y > 0, 1 f(y) = yσ y µ) /σ e (ln ; π therefore, the relationship P (a Y b) = P ( ln a X ln b ) allows lognormal probabilities for Y to be computed using normal probabilities for X 7

8 lognormal parameters where Y is lognormal (i.e., X = ln Y is normal having mean µ and standard deviation σ), then Y = e X and E(Y ) = µ Y = e µ+1 σ SD(Y ) = σ Y = (e σ 1)e µ+σ where ( ) µ Y µ = ln µ Y + σy and σ = ln ( 1 + σ Y µ Y ) 8