Comp Sci
Uploaded by blackboi66 on Nov 30, 2005
Random Variable
A random variable X is a rule that assigns a numerical value to each outcome in
the sample space of an experiment.
A discrete random variable can take on specific, isolated numerical values, like
the outcome of a roll of a die, or the number of dollars in a randomly chosen
bank account.
A continuous random variable can take on any values within a continuum or an
interval, like the temperature in Central Park, or the height of an athlete in
centimeters.
Discrete random variables that can take on only finitely many values (like the
outcome of a roll of a die) are called finite random variables.
Probability Distribution
The probability P(X = x) is the probability of the event that X = x. Similarly,
the probability that P(a < X < b) is the probability of the event that X lies
between a and b.
These probabilities may be estimated, empirical, or abstract
For a finite random variable, the collection of numbers P(X = x) as x varies is
called the probability distribution of X, and it is useful to graph the
probability distribution as a histogram.
Bernoulli Trials and the Binomial Distribution
A Bernoulli trial is an experiment with two possible outcomes, called success
and failure. Each outcome has a specified probability: p for success and q for
failure (so that p+q = 1).
If we perform a sequence of n independent Bernoulli trials, then some of them
result in success and the rest of them in failure. The probability of exactly x
successes in such a sequence is given by
P(exactly x successes in n trials) = C(n,x)pxqn-x.
If X is the number of successes in a sequence of n independent Bernoulli trials,
with probability p for success and q for failure, then X is said to have a
binomial distribution. This distribution is given by the above formula
P(X = x) = C(n,x)pxqn-x
for x running from 0 to n.
Measures of Central Tendency:
Mean, Median, and Mode of a Set of Data
A collection of specific values, or "scores", x1, x2, . . ., xn of a random
variable X is called a sample. If {x1, x2, . . ., xn} is a sample, then the
sample mean of the collection is
x = x1 + x2 + . . .+ xn
n
= xi
n ,
where n is the sample size: the number of scores. The sample median m is the
middle score (in the case of an odd-size sample), or average of the two middle
scores (in the case of an even-size sample), when the scores in a sample are
arranged in ascending order.
A sample mode is...