4.1 Probability Distributions
• Important Concepts
– Random Variables
– Probability Distribution
– Mean (or Expected Value) of a Random
Variable
– Variance and Standard Deviation of a
Random Variable
4.1 Probability Distributions
• Consider the following experiment:
– Suppose we toss a coin three times.
• What is the sample space for this experiment?
• What is the probability of tossing exactly:
0 tails?
1 tails?
2 tails?
3 tails?
4.1 Probability Distributions
• Terms to know:
– A random variable X represents a numerical
value associated with each outcome of a
probability experiment.
• A random variable is discrete if it has a finite or
countable number of possible outcomes.
• A random variable is continuous if it has an
uncountable number of possible outcomes.
4.1 Probability Distributions
• Terms to know:
– A discrete probability distribution lists each
possible value a random variable can
assume, together with its probability.
• All probability distributions must satisfy the
following two conditions:
– The probability of each value of the random variable
must be between 0 and 1, inclusive.
– The sum of all the probabilities must be 1.
#26 p. 198
#28 p. 198
4.1 Probability Distributions
• Discrete probability distribution of our
random variable X:
# of tails, x
P( X = x )
0
1/8 = 0.125
1
3/8 = 0.375
2
3/8 = 0.375
3
1/8 = 0.125
1.000
4.1 Probability Distributions
• How do we find the mean and standard
deviation of a discrete random variable?
In chapter 3, we used the following:

x
N

x  
2 
 x   
2
N
N
Can we still use these formulas?

2

2
x

N
2
x

N
 2
 2
4.1 Probability Distributions
• Let’s try #31 p. 199 (Camping Chairs)
Number of
Cats, x
P(X=x)
x·P(X=x)
x2
x2·P(X=x)
0
0.250
0
0
0
1
0.298
0.298
1
0.298
2
0.229
0.458
4
0.916
3
0.168
0.504
9
1.512
4
0.034
0.136
16
0.544
5
0.021
0.105
25
0.525
1.501
3.795
4.1 Probability Distributions
• #33 p. 199 (Hurricanes)
Category, x
P(X = x)
x∙P(X =x)
x2
x2∙P(X =x)
1
0.418
0.418
1
0.418
2
0.261
0.522
4
1.044
3
0.247
0.741
9
2.223
4
0.063
0.252
16
1.008
5
0.010
0.05
25
0.25
1.983
4.943