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If we can reduce our desire, then all worries that bother us will disappear. Random Variables and Distributions. Distribution of a random variable Binomial and Poisson distributions Normal distributions. What Is a Random Variable?.
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If we can reduce our desire, then all worries that bother us will disappear.
Random Variables and Distributions Distribution of a random variable Binomial and Poisson distributions Normal distributions
What Is a Random Variable? • The numerical outcome of a random circumstance is called a random variable. Eg. Toss a dice: {1,2,3,4,5,6} Height of a student • A random variable (r.v.) assigns a number to each outcome of a random circumstance. Eg. Flip two coins: the # of heads
Types of Random Variables • A continuous random variable can take any value in one or more intervals. ** eg. Height, weight, age • A discrete random variable can take one of a countable list of distinct values. ** eg. # of courses currently taking
Distribution of a Discrete R.V. • X = a discrete r.v. • x = a number X can take • The probability distribution function (pdf) of X is: P(X = x)
Example: Birth Order of Children ** pdf: Table 7.1 on page 163 ** histogram of pdf: Figure 7.1
Important Features of a Distribution • Overall pattern • Central tendency – mean • Dispersion – variance or standard deviation
Calculating Mean Value • X = a discrete r.v. • { x1, x2, …} = all possible X values • pi is the probability X = xi where i = 1, 2, … • The mean of X is:
Variance & Standard Deviation • Notations as before • Variance of X: • Standard deviation (sd) of X:
Bernoulli and Binomial Distributions • A Bernoulli trial is a trial of a random experiment that has only two possible outcomes: Success (S) and Failure (F). The notational convention is to let p = P(S). • Consider a fixed number n of identical (same P(S)), independent Bernoulli trials and let X be the number of successes in the n trials. Then X is called a binomial radon variable and its distribution is called a Binomial distribution with parameters n and p. Read the handout for bernoulli and binomial distributions.
PDF of a Binomial R.V. • p = the probability of success in a trial • n = the # of trials repeated independently • X = the # of successes in the n trials For x = 0, 1, 2, …,n, P(X=x) =
Mean & Variance of a Binomial R.V. • Notations as before • Mean is • Variance is
Brief Minitab Instructions • Minitab: Calc>> Probability Distributions>> Binomial; Click ‘probability’ , ‘input constant’ and n, p, x • Minitab Output: Binomial with n = 3 and p = 0.29 x P( X = x ) 2 0.179133
The Poisson Distribution • a popular model for discrete events that occur rarely in time or space such as vehicle accident in a year • The binomial r.v. X with tiny p and large n is approximately a Poisson r.v.; for example, X = the number of US drivers involved in a car accident in 2008 Read the Poisson distribution handout.
Brief Minitab Instructions • Minitab: Calc>> Probability Distributions>> Poisson; Click ‘probability’ , ‘input constant’ and l,x • Minitab Output: Poisson with mean = 2.4 x P( X = x ) 1 0.217723
Distribution of a Continuous R.V. • The probability density function (pdf) for a continuous r.v. X is a curve such that P(a < X <b) = the area under it over the interval [a,b].
Normal Distribution • Its density curve is bell-shaped • The distribution of a binomial r.v. with n=∞ • The distribution of a Poisson r.v. with l=∞ Read the normal distribution handout.
Standard Normal Distribution • X: a normal r.v. with mean m and standard deviation s • Thenis a normal r.v. with mean 0 and standard deviation 1; called a standard normal r.v.
Brief Minitab Instructions • Minitab: Calc>> Probability Distributions>> Normal; Click what are needed • Minitab Output: Inverse Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1 P( X <= x ) x 0.95 1.64485 Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1 x P( X <= x ) 1.64485 0.950000
Example: Systolic Blood Pressure • Let X be the systolic blood pressure. For the population of 18 to 74 year old males in US, X has a normal distribution with m = 129 mm Hg and s = 19.8 mm Hg. • What is the proportion of men in the population with systolic blood pressures greater than 150 mm Hg? • What is the 95-percentile of systolic blood pressure in the population?