23-07-2014, 10:23 AM
Probability Distributions
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Distributions
A random variable is a quantitative variable whose value depends on chance.
We’ll use them to model the behavior of populations.
A discrete random variable is a random variable whose possible values form
a finite or countably infinite set of numbers.
A continuous random variable is a random variable whose possible values
form a continuum or interval of numbers.
Use capital letters for random variables and lower case letters for their
possible values.
The distribution of a random variable is TWO things:
1. a list of the possible values it can take on;
2. a list of the probabilities of those possible values.
Mean of a Random Variable
The mean is a weighted average of the possible values of x, where the
weights are the probabilities.
If the distribution of X describes a population, the sample means of large
samples tend to be close to the mean µ
Binomial Random Variables
X is a binomial random variable with parameters n and p if
P(X = x) =
n
x
p
x
(1 − p)
n−x
where x is any number in {0, 1, 2, . . . , n}. X denotes the number of successes
that occur in n independent trials where each trial has success probability p.
Example 4 There are 49 senators, each of whom shows for the senate
meeting with probability 0.95. What is the probability that no one shows?
What is the probability that everyone shows? What is the probability that
exactly 46 show? What is the probability that at least 47 show?
More about Normal Random Variables
Normal random variables arise in the study of natural and industrial systems.
Many statistical tests are based on normal distributions.
If X has a normal distribution with mean µ and standard deviation σ, then
P(a ≤ X ≤ b) is the area of the region under the normal curve between the
points a and b above the x-axis: