2.2: continuous random variables and probability distributions

shape of normal distribution

changing mu (mean) shifts the distribution left or right; changing omega (standard dev) increases/decreases spread X ~ N (mu, omega squared)

Finding the X value

convert to X units using the formula X = mu + Z standard dev

marginal distribution fxn

cumulative distribution functions of independent random variables

Normal Probability Density Function

describes a symmetric, bell shaped curve; completely defined by the mean and variance (standard deviation) omega squared is the pop variance, mu is the pop mean, x is any value of the continuous variable

the uniform distribution

equal probabilities for all equal-width intervals within the range of the random variable

normal distribution approx for binomial distribution characteristics

shape of binomial distrib is approx normal if n is large; normal is a good approx to the binomial when nP (1-P) larger then 5 let X be the number of successes from n independent trials, each w probability of success P

finding normal probabilities example

suppose X is normal with mean 8 and standard dev 5.0. Find P (x smaller than 8.6); equation is 8.6-8/5 = .12 = z; then, put into a standardized normal prob table, and the F(z) value is .54

shaded area under the curve is...

the probability that X is between a and b

probability as an area

the total area under the curve f(x) is 1; the area under the curve f(x) to the left of x0 is F(x0) where x0 is any value that the random value can take

procedure for finding probabilities

to find P between a and b when X is distributed normally: draw the normal curve for the problem in terms of X, translate X-values to Z-values, use cumulative normal table

probability as area under the curve

total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below (half on left of mu is .5, half right of mu is .5, whole curve is 1.0)

continuous random variable

variable that can assume any value in an interval (ex: thickness of an item, time req to complete a task, temp of a solution, height in inches)

standard normal distribution table

F(a) = P(Z smaller than a); found in Appendix Table 1

finding the X value for a known probability

1. find the Z value for a known probability 2. convert to X units using the formula (X = mu + Z omega)

Exponential Distribution

A probability distribution associated with the time between arrivals cumulative distribution fxn (prob that an arrival time is less than the specified time t) is F(t) = 1- e to the -lambda t

cumulative distribution function

F(x), for a cont random variable X expresses the probability that X does not exceed the value of x the area under the prob density function f(x) from the minimum x value up to x0

correlation btwn x and y

Corr (X,Y) = Cov (X,Y) / omega X omega Y

Cov (x,y)

E [ (X- mu x) (Y- mu y) ]

cumulative normal distribution

The probability that an outcome from a standard normal distribution will be below a certain value. F(x zero) = P (X smaller than or equal to x 0)

jointly distributed continuous random variables

X1, X2, etc cont random variables; their joint cumulative distribution fxn is F(x1, x2, etc) and defines prob that X1 less than x1, X2 less than x2, so on

finding normal probabilities

probability for a range of values is measured by the area under the curve

standard normal distribution

any normal distribution (w any mean and variance combo) can be transformed into the standardized normal distribution (Z), with mean 0 and variance 1

the normal distribution

bell shaped, symmetrical, mean/median/mode all equal; location determined by mean (mu), spread determined by standard dev (omega); range theoretically infinite

normal distribution approx for binomial distribution

binomial (n independent trials, prob of success on any given trial = P); random variable X: Xi = 1 if the ith trial is "success"; Xi=0 if the ith trial is failure

probability density function

f(x) or a random variable X has properties: 1.) f(x) is bigger than 0 for all values x, 2.) area under the probability density fxn f(x) over all values of random var X within range is equal to 1.0 3.) prob that X lies between two values (a and b) is the area under the density fxn graph between the two values

financial portfolio

linear combo of separate financial instruments (proportion of port value in stock 1) x (stock 1 return) etc

standardized random variable

mean of 0, variance 1 Z = X - mu x / omega x

expectations for continuous random variables

mean of X, denoted mu x, is defined as the expected value of X (E[X]) variance of X, denoted omega x squared, is defined as the expectation (E[..]) of the squared dev, (X - mu x) squared, of a random variable from its mean

sums of random variables

mean of their sum is the sum of their means

mean/variance of uniform distribution

mean- mu is a+b/2 variance- omega = (b-a) squared /12 where a= min value of x, b= max value of x

assessing normality

not all continuous random variables are normally distributed; its important to eval how well the data is approx by a normal distribution