Probability Distributions

Random Variables * Discrete Probability Distributions * Mean, Variance, and

Standard Deviation * Expected Value

Fill in the blanks 1 – 3

1. Random Variable: A count or an ________________ of a probability experiment. (Uses numerical values)

outcome

2. Discrete random variable: Has a finite or __________________ number of possible outcomes that can be listed.

countable

3. Continuous random variable: Has a _____________ number of possible outcomes, represented by an interval on the number line.

noncountable

Problem # 44. Discrete probability distribution: Lists each _____________

value that the random variable can assume, together with its probability. The following conditions must apply:

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

                                              i.      ________________________

b. The sum of all the probabilities is 1.

                    i.      = _________________)(xxP

{0, 1}

possible

1

5 - 8

5. Mean of a discrete random variable: Each probability of x is __________ by its corresponding probability and the products are added.

*Mean values may be bigger than 1 since they are for more than one person!

)(xxPmultiplied

6. Variance of a discrete random variable is

)(22 xPx

7. Standard Deviation is 2

8. Expected Value: A ____________ random variable is equal to the mean of the random variable.Expected value = (note that the mean and expected values are the same!)

discrete

)()( xxPxE

Example #1

x 0 1 2 3 4 5

P(x) 0.05 0.23 0.10 ? 0.11 0.16

Determine the missing probability value

0.035

Example #2

Score, x Frequency, f Relative frequency P(x)

1 30 0.21

2 31

3 12 0.09

4 28

5 39 0.28

Constructing a Discrete Probability Distribution

1. Make a frequency distribution for the possible outcomes.

2. Find the sum of the frequencies.

3. Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies.

4. Check that each probability is between 0 and 1 and that the sum is 1.

0.22

0.20

______f 140 ______)( xP 1

Example #3Example #3 Is the following a probability distribution? ______________

x 0 1 2 3 4

P(x) 0.40 0.10 0.25 0.30 .05

NO! it adds up to 1.1

Example #4

Dogs 0 1 2 3 4 5

Households 1491 425 168 48 29 14

Hint: Use the lists in your calculator!

Example #4 Continued

More Example #4

c) Find the variance ___________________

d) Find the standard deviation _____________________

b) Find the mean _______________________

e) A household on average has _________ dogs with a standard deviation of ________________.

0.50

0.83

0.9

0.5*0.9

Note, if it is a nice dog I want the front half!

Example #5A charity organization is selling $4 raffle tickets part of a fund-raising program. The first prize is a boat valued at $3150, and second prize is a camping tent valued at $450. The rest of the prizes are 15 - $25 gift certificates. The number of tickets sold is 5000. Find the expected net gain to the player for one play of the game. Is the players expected to win or to lose?