tree diagrams and Venn diagrams are other useful ways to organize data, and
calculate conditional probability, particularly if there are many cases to
consider
4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
recommended:
1-17, 69
remember:
discrete data are data that you can count
a discrete random variable \( X \) describes the outcome of a statistical
experiment in words
the value \( x \) of a discrete random variable is a number
a discrete probability distribution function, \( P(x) \), is a function
with two characteristics
each probability (value of \( P(x) \)) is between \( 0 \) and \( 1 \)
inclusive
\( 0 \le P(x) \le 1\), for all \( x\in X \)
the sum of the probabilities (the values of \( P(x) \)) is \( 1 \)
\( \sum_{x\in X} P(x)=1 \)
4.2 Mean or Expected Value and Standard Deviation
recommended:
18-36, 70-81
the standard deviation of a discrete random variable
calculation
multiply the square of the deviation of each value of the random
variable by the probability of that value, add the products, and
take the square root of that value
the expected value is sometimes referred to as the
long-term average or mean, denoted by the Greek letter \( \mu \)
calculation
multiply each value of the random variable by its probability and add
these products
\( \mu = \sum_{x\in X} xP(x) \)
the expected value is sometimes referred to as the
long-term average or mean, denoted by the Greek letter \( \mu \)
the Law of Large Numbers states that as the number of trials in a
probability experiment increases, the difference between the theoretical
probability of an event and the relative frequency of the event approaches zero
4.3 Binomial Distribution
recommended:
37-44, 82-103
a binomial experiment has three characteristics
there are a fixed number of trials
there are only two possible outcomes
the trials are independent and repeated under identical conditions
notation
\( n \), the number of trials
\( p \), the probability of "success"
\( q \), the probability of "failure", \( q=1-p \)
\( X\mbox{~}B(n,p) \), "\( X \) is a random variable with a binomial probability
distribution of \( n \) trial snd \( p \) probability of success"
the mean and standard deviation for a binomial probability distribution \( X\mbox{~}B(n,p) \)
are \( \mu = np \) and \( \sigma = \sqrt{npq} \)
an individual trial of a binomial experiment is called a Bernoulli trial
for a binomial probability distribution \( X\mbox{~}B(n,p) \), the probability of
\( x \) successes in \( n \) trials, where the probability of success is \( p \)
and the probability of failure is \( q=1-p \) is given by formula
\( P(x)=C(n,x)p^{x}q^{n-x} \), where \( \displaystyle{C(n,x)=\frac{n!}{x!(n-x)!}} \)
is called the binomial coefficient and \( n!=n(n-1)\cdots 2\cdot 1 \)
the product of the positive integers from \( n \) to \( 1 \)