Download Probability Distributions

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 ________________
outcome
of a probability experiment. (Uses numerical values)
2. Discrete random variable: Has a finite or
countable
__________________
number of possible outcomes that
can be listed.
noncountable
3. Continuous random variable: Has a _____________
number of possible outcomes, represented by an
interval on the number line.
Problem # 4
possible
4. 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.
{0, 1}
i. ________________________
b. The sum of all the probabilities is 1.
1
i. xP(x ) = _________________
5-8
5. Mean of a discrete random variable:   xP(x)
multiplied by its corresponding
Each probability of x is __________
probability and the products are added.
*Mean values may be bigger than 1 since they are for more
than one person!
6. Variance of a discrete random variable is  2  x   2 P( x)
2
7. Standard Deviation is   
8. Expected Value: A ____________
random variable is equal
discrete
to the mean of the random variable.
Expected value = E ( x)    xP( x)
(note that the mean
and expected values are the same!)
Example #1
Determine the missing probability value
x
P(x)
0
0.05
1
0.23
2
0.10
3
4
?0.035 0.11
5
0.16
Example #2
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.
Score, x
Frequency, f
Relative frequency P(x)
1
30
0.21
2
31
0.22
3
12
0.09
4
28
0.20
5
39
0.28
f  ______
140
P( x)  ______
1
Example #3
NO! it adds
Example #3 Is the following a probability distribution? ______________
up to 1.1
x
0
1
2
3
4
P(x)
0.40
0.10
0.25
0.30
.05
Example #4
Dogs
0
Households 1491
1
2
3
4
5
425
168
48
29
14
Hint: Use the lists in your calculator!
Example #4 Continued
More Example #4
0.50
b) Find the mean _______________________
c) Find the variance ___________________
0.83
0.9
d) Find the standard deviation _____________________
e) A household on average has _________
0.5*
dogs with a
standard deviation of ________________.
0.9
Note, if it is a nice dog I want the front half!
Example #5
A 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?