Binomial & Geometric

Random Variables

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Goals:Binomial settings and binomial random variables

Binomial probabilitiesMean and standard deviation of a binomial distribution

Binomial distributions in statistical sampling

Geometric random variables

What do these have in common?

Toss a coin 5 times. Count the # of heads.Spin a roulette wheel 8 times. Record how

many time the ball lands in a red slotTake a random sample of 100 babies born in

the US today. Count the number of little girls.

Repeated trials of the same chance process# of trials is fixed in advanceTrials are independentLooking for a # of successes Chance of success is the same for each trial

BS…Binomial SettingWhen these conditions are meet we have a binomial setting.

Definition A binomial setting arises when we

perform several independent trials of the same chance process and record the number of items that a particular outcome occurs.

The 4 conditions for a binomial setting are Binary? Independent? Number? Success?

BINS

“BINS” Binary…possible outcomes can be classified as a “success” or “failure” Independent…the result of one trial cannot have an effect on another trial Number…the # of trials, n, is fixed in advance Success…probability of success on each trial is the same

Binomial random variable & binomial distribution

The count X of successes in a binomial setting is a binomial random variable.

The probability distribution of X is a binomial distribution with parameters n and p, where n is the # of trials of the chance process and p is the probability of a success on any one trial.

The possible values of X are the whole numbers from 0 to n.

ExamplesType O blood….Turn over 10 cards record aces…Turn over top card, replace, repeat until …

More examples1. Shuffle a deck of cards. Turn over the top

card. Put the card back in the deck, and shuffle again. Repeat this process 10 times. Let X = the # of aces you observe.

2. Choose three students at random from your class. Let Y = the # who are over 6 feet tall.

3. Flip a coin. If it’s heads, roll a 6-sided die. If it’s tails, roll and 8-sided die. Repeat this process 5 times. Let W = the # of 5’s you roll.

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