**Random**

Variables

Variables

**Discrete Probability Distributions**

**Finding mean, variance, and standard deviation of probability distribution**

Random variable x represents a numerical value associated with each outcome of a probability experiment.

The mean of a discrete random variable is given by

The mean of a random variable represents what you would expect to happen for thousands of trails. It is also called the "expected value".

**Probability Distributions**

**WRite Down EVERYTHING!**

Now you try...

Decide whether the random variable x is discrete or continuous.

Any questions?

You can use a relative frequency distribution histogram to represent this type of distribution because probabilities represent relative frequencies.

A random variable is either discrete or continuous.

variable is discrete if it has a finite or countable number of possible outcomes that can be listed

variable is continuous if it has an uncountable number of possible outcomes, represented by an interval on the number line.

Number of sales calls a salesperson can make in one day is discrete because the set of possible outcomes can be listed.

These can only be whole numbers.

Time in hours a salesperson spends making calls in one day is continuous because the set of possible outcomes can be any number from 0 to 24 including fractions and decimals.

These can be an interval on a number line, but you CANNOT list all the values.

x represents the volume of water in a 32-ounce container.

A discrete probability distribution lists each possible value the random variable can assume, together with its probability.

A probability distribution must satisfy the following conditions:

The probability of each value of the discrete random variable is between 0 and 1, inclusive.

The sum of all the probabilities is 1.

Lets make our own

discrete probability!

An industrial psychologist administered a personality inventory test for passive-aggressive traits to 150 employees. Individuals were given a score from 1 to 5, where 1 was extremely passive and 5 extremely aggressive. A score of 3 indicated neither trait. The results are shown in the next table. Construct a probability distribution for the random variable x. Then graph the distribution using a histogram.

Score, x

Frequency, P(x)

1

2

3

4

5

24

33

42

30

21

1st find the probability of each value, i.e. find the relative frequency.

2nd create the relative frequency histogram. Note that relative frequency is always the y axis.

Verifying Probability Distributions

The distribution must satisfy these two conditions:

Each probability is between 0 and 1.

The sum of the probabilities equals 1.

Decide whether each distribution is a probability distribution.

x

P(x)

5

0.28

6

0.21

7

0.43

8

0.15

x

P(x)

1

1/2

2

1/4

3

5/4

4

-1

Now find the variance and standard deviation of the previous data.

Although probabilities can never be negative, the expected value of a random variable can be negative.

At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy one ticket. What is the expected value of your gain?

Gain, x

P(x)

Binomial experiment is a probability experiment that satisfies the following conditions.

The experiment is repeated for a fixed number of trials, where each trial is independent of the other trials.

There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success (S) or as a failure (F).

The probability of a success P(S) is the same for each trial.

The random variable x counts the number of successful trials.

Binomial notations are as follows:

n = number of times a trial is repeated

p = probability of success in a single trial

q = The probability of failure in a single trial (q = 1 - p)

x = the random variable represents a count of the number of successes in n trials: x= 0, 1, 2, 3,...,n

You pick a card from a standard deck of cards and note whether it is a club or not and then place the card back into the deck. You repeat the experiment five times, so n = 5. The outcomes for each trial can be classified into two categories: S = selecting a club and F = selecting another suit. Then if x = 2, then exactly two of the five cards are clubs and the other three are not clubs.

Decide whether the experiment is a binomial experiment.

A jar contains five red marbles, nine blue marbles, and six green marbles. You randomly select three marbles from the jar, without replacement. The random variable represents the number of red marbles.

Binomial Probability Formula

Microfracture knee surgery has a 75% chance of success on patients with degenerative knees. The surgery is performed on three patients. Find the probability of the surgery being successful on exactly two patients.

n

p

q

x

3

3/4

1/4

2

Mean, Variance, and Standard Deviation of Binomial Distribution

Mean = np

Variance = npq

Standard Deviation = square root of (npq)

Geometric Distribution

A geometric distribution is a discrete probability distribution of a random variable x that satisfies the following conditions.

A trial is repeated until a success occurs.

The repeated trials are independent of each other.

The probability of success p is constant for each trial.

The probability that the first success will occur on trial number x is P(x) = p(q)^x-1, where q = 1-p

From experience, you know that the probability that you will make a sale on any given telephone call is 0.23. Find the probability that your first sale on any given day will occur on your fourth or fifth sales call.

p

q

x

Poisson Distribution

The Poisson distribution is a discrete probability distribution of a random variable x that satisfies the following conditions:

The experiment consists of counting the number of times, x, an event occurs in a given interval. The interval can be an interval of time, area, or volume.

The probability of the event occurring is the same for each interval

The number of occurrences in one interval is independent of the number of occurrences in other intervals.

The probability of exactly x occurrences in an interval is

The mean number of accidents per month at a certain intersection is three. What is the probability that in any given month four accidents will occur at this intersection?

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