probability distributions random variable: represents a numerical value associated with each outcome...

Probability Distributions

Random Variable: Represents a numerical value associated with each outcome of a probability experiment:

Example: Flip a coin three times:

Let x be number of heads, then

x = 0, 1, 2, 3

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Types of Data

Quantitative variables are either discrete or continuous.

Discrete: Counts. Must “jump” from one data point to the next.

Continuous: Measures: Can be made more and more precise.


A Discrete Probability Distribution lists each value for the random variable and its associated probability. All probabilities sum to 1.

Example: Toss a coin three times

X P(x)

0 1/8

1 3/8

2 3/8

3 1/8

total 1

X = number of heads

Sample Space:HHH, HTT, THT, TTHTHH, HTH, HHT, TTT


Example: Toss a coin three times

X P(x)

0 1/8

1 3/8

2 3/8

3 1/8

total 1

X = number of heads

Sample Space:HHH, HTT, THT, TTHTHH, HTH, HHT, TTT

1/8

3/8 3/8

1/8

0 1 2 3


Example: Is this a probability distribution?

No, because.28+.21+.43+.15 <>1

X P(x)

5 .28

6 .21

7 .43

8 .15

total 1.07


Example: Is this a probability distribution?

Yes, because.28+.21+.43+.08 =1

X P(x)

5 .28

6 .21

7 .43

8 .08

total 1


Example: Make this a a probability distribution.

If the probability for 7 is .36, then all probabilities add to one.

X P(x)

5 .28

6 .21

7 ?

8 .15

total 1


Find the mean of a probability distribution:

Calculate the same as the mean of a frequency distribution

X P(x) X*p(x)

5 .28 1.4

6 .21 1.26

7 .36 2.52

8 .15 1.2

total 1 E(x) = 6.38


Find the standard deviation of a probability distribution:

Var(x) = E(x^2) – E(x)^2 = 41.8 – 6.382 = 1.096SD = sqrt(1.096) = 1.05Calculate the same as the standard deviation of a frequency distribution

X P(x) X*p(x) X^2*p(x)

5 .28 1.4 7

6 .21 1.26 7.56

7 .36 2.52 17.64

8 .15 1.2 9.6

total 1 E(x) = 6.38 E(x^2) =41.8


What is the expected value and SD for rolling a pair of dice?X P(x) X*P(x) X^2*p(x)

2 1/36 2/36 4/36

3 2/36 6/36 18/36

4 3/36 12/36 48/36

5 4/36 20/36 100/36

6 5/36 30/36 180/36

7 6/36 42/36 294/36

8 5/36 40/36 320/36

9 4/36 36/36 324/36

10 3/36 30/36 300/36

11 2/36 22/36 242/36

12 1/36 12/36 144/36

Total 36/36 = 1 256/36=7 1974/36

=54.833

Expected value =7variance = 54.833-72 = 5.833Standard deviation= sqrt(5.833)=2.42


Probability function:Is P(x) =(x+2)/14 for x = 0, 1, 2, 3 a probability function?

yes! All probabilites<1 and sum to 1

X P(x)

0 2/14

1 3/14

2 4/14

3 5/14

1


Probability function:Is P(x) =(x+2)/10 for x = 0, 1, 2, 3 a probability function?

No! All probabilities do not sum to 1

X P(x)

0 2/10

1 3/10

2 4/10

3 5/10

14/10


You are playing Deal or No Deal. There are three cases left plus your case the amounts left are $1; $50; $10,000; and $500,000.

a) What is the expected value?b) The banker makes you an offer of $100,000. Is this more or

less than the expected value?X P(x)

1 .25

50 .25

10,000 .25

500,000 .25

1

c) If you could play and infinite number of games, the best strategy would be to never take an offer less than the expected value and to always take an offer more than the expected value. Why might this strategy differ for a single game?


You are playing Deal or No Deal. There are three cases left plus your case the amounts left are $1; $50; $10,000; and $500,000.

a) What is the expected value?b) The banker makes you an offer of $100,000. Is this more or

less than the expected value?X P(x) X*P(x)

1 .25 .25

50 .25 50/4

10,000 .25 10000/4

500,000 .25 500000/4

1 127512.75

c) If you could play and infinite number of games, the best strategy would be to never take an offer less than the expected value and to always take an offer more than the expected value. Why might this strategy differ for a single game?

probability distributions random variable: represents a numerical value associated with each outcome...

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