CH 6 Notes

The probability distribution of a discrete random variable X provides the possible values of the random variable and their corresponding probabilities. A probability distribution can be in the form of a​ table, graph, or mathematical formula.

Let​ P(x) denote the probability that the random variable X equals​ x, then

ΣP(x) = 1 and 0 ≤ P (x) ≤1.

What are the two requirements for a discrete probability​ distribution?

ΣP(x) = 1 and 0 ≤ P (x) ≤1.

As the number of experiments​ increases, the mean of the observations will approach the mean of the random variable.

Criteria for a Binomial Probability Experiment

An experiment is said to be a binomial experiment if

1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial.

2. The trials are independent. This means that the outcome of one trial will not affect the outcome of the other trials.

3. For each trial, there are two mutually exclusive (disjoint) outcomes: success or failure.

4. The probability of success is the same for each trial of the experiment.

Notation Used in the Binomial Probability Distribution

• There are n independent trials of the experiment.

• Let p denote the probability of success for each trial so that 1−p is the probability of failure for each trial.

• Let X denote the number of successes in n independent trials of the experiment. So 0 ≤ x≤ n .

The formula for the expected number of successes in a binomial experiment with n trials and probability of success p is np.

What is the formula for the expected number of successes in a binomial experiment with n trials and probability of success​ p? E(X) = np

For a fixed​ p, as the number of trials n in a binomial experiment​ increases, the probability distribution of the random variable X becomes more bell shaped.

As the number of experiments​ increases, the mean of the observations will approach the mean of the random variable.