Binomial Random Variables
If X has the binomial distribution with n observations and probability of success, p, on each observation, then the possible values of X are what?
0, 1, 2, ..., n with k being any one of these values.
Binomial distributions are important to statistics when we are trying to do what?
when we are trying to make inferences about the proportion, p, of successes in a population.
A binomial distribution has three important pieces:
x = the number of successes. n = the number of trials. p = the probability of success.
Mean & Standard Deviation of a Binomial Random Variable: If a count X has the binomial distribution with number of observations, n, and probability of success, p, the mean and standard deviation of X are what?
Look in notes.
We have what to simplify this process?
Luckily, we have the Binomial Probability Formula which simplifies this process.
There are four conditions that must be met for a binomial random variable:
1.There are two possible outcomes, success or failure. 2.There are a fixed number of trials/observations. 3.Each trial/observation is independent. 4.The probability of success, p, is the same for each trial/observation.
As an abbreviation, we say that a random variable X is what?
B(n, p).
The Binomial Coefficient:
The number of ways of arranging k successes among n observations is given by: The binomial coefficient "n choose k".
The possible values of X are what?
The possible values of X are the whole numbers from 0 to n, the number of trials.
When the population is much larger than the sample, a count of successes in an SRS of size n has approximately the binomial distribution with what?
has approximately the binomial distribution with n equal to the sample size and p equal to the proportion of successes in the population.
As you can see, this can be tedious if n is what?
if n is large and it is quite possible to miss some combinations.
success does not always imply something positive, it is simply what?
the desired outcome for X.
In order to computing probabilities for binomial experiments, we can use what?
we can use the rules of probability that we have already learned.