7.1 Discrete and Continuous Random Variable
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Calculate the probability of a discrete random
variable and display in a graph.
Calculate the probability of a continuous
random variable using a density curve.
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We toss a coin 4 times. Our outcome is
HTTH.
Let x=# of heads, therefore x=2.
If we got TTTH, then x=1.
The values of x are 0,1,2,3,4. (x is called a
random variable)
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RANDOM VARIABLE:
a variable whose value is a numerical
outcome of a random phenomenon
DISCRETE RANDOM VARIABLE X:
has a countable # of outcomes (possible
values)
Lists their values and their probabilities.
Value of X
x₁
x₂
x₃
....
Probability
p₁
p₂
p₃
....
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Two Requirements that the probabilities must satisfy
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1- p has to be between 0 and 1.
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2- p₁ + p₂ + …
=1
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Ex. 1: The instructor of a large class gives 15% each of
A’s and D’s, 30% each of B’s and C’s, and 10% F’s.
Choose a student at random from this class. To
“choose at random” means to give every student the
same chance to be chosen. The student’s grade on a
four point scale (A=4) is a random variable x:
Grade
0
1
2
3
4
Prob.
0.10
0.15
0.30
0.30
0.15
P(Grade is a B or higher) =
P(3 or 4)= P(3) +P(4) = 0.3 + 0.15=0.45
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used to picture the probability distributions
of a discrete random variable.
Create a histogram for the above data:
Grades
0.4
0.3
0.2
Grades
0.1
0
0
1
2
3
4
We Know:
1- outcomes ( H or T)
2- independent
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There are 16 possible outcomes.
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For ex: P(HTTH)= 1/16
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What is P(x=0)= 1/16 (TTTT)
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P(x=1)= 1/4
P(x=3)= 1/4
P(x=2)=3/8
P(x=4) = 1/16
Number of
heads
0
1
2
3
4
Probability
1/16
1/4
3/8
1/4
1/16
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What is the probability of tossing at least 1 head?
P(x≥1)=1-P(0)= 1- (1/16)= 15/16
What is the probability of tossing no more than 3
heads?
P(x≤3)= P(x<4)= 1- P(x=4)=
1-(1/16)=15/16
A) 1%
 B) All probabilities are between o and 1. They
also add up to 1.
 C) P(x≤3)= 0.94
 D) P(x<3)=0.86
 E) P(x≥4)=P(x>3)= 0.06
 F)Let 01-48=class 1 ,49-86=class2
87-94=class 3, 95-99=class 4, 00= class 5.
Use a RDT to repeatedly generate 2 digit #’s to
find the proportion of those from 01-94.
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Look at Figure 7.4 pg. 375 (spinner)
A spinner generates a random number between 0 and
1.
What is the sample space?
S={ 0 ≤ x < 1 }
*We cannot assign probabilities to each individual
value of x and then sum, because there are infinitely
many possible values.
-Instead we use intervals (area under a density
curve)!!! (A new way of assigning probabilities
directly to events).
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What is P(.3 < x < .7) = .4
P(x<.5)= 0.5
P(x > .8)= 0.2
P(x < .5 or x > .8)= 0.7
P(x = .8)= 0
We call X a continuous random variable
because its values are not isolated #’s but an
entire interval of #’s
CONTINUOUS RANDOM VARIABLE Xtakes all values in an interval of numbers
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PROBABILITY DISTRIBUTIONdensity curve
Difference between the two random variables:
discrete (specific values)
continuous (intervals)
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N ( μ , σ ) = N(mean, standard deviation)
The standardized variable is
Z= (x-µ)/σ
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What is P(p< .28 or p> .32)
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Step 1: Draw and shade your normal curve:
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Step 2: Convert to z-scores
P( p< .28)= P(z< (0.28-0.3)/0.0118)=
P(z<-1.69)= 0.0455
P(p > .32) =P(z> (0.32-0.3)/0.0118)=
P(z>1.69)= 0.0455
0.0455+0.0455= 0.091