Question 1182767
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Part (a)


Given table<table border = "1" cellpadding = "5"><tr><td>X</td><td>70</td><td>80</td><td>90</td></tr><tr><td>P(X)</td><td>0.3</td><td>0.45</td><td>0.25</td></tr></table>


Form a new row that consists of multiplying the X and P(X) values
Example: 70*0.3 = 21 in the first column
<table border = "1" cellpadding = "5"><tr><td>X</td><td>70</td><td>80</td><td>90</td></tr><tr><td>P(X)</td><td>0.3</td><td>0.45</td><td>0.25</td></tr><tr><td>X*P(X)</td><td>21</td><td>36</td><td>22.5</td></tr></table>
Add up everything in that third row: 21+36+22.5 = 79.5
This is the expected value.


Answer: 79.5


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Part (b)


Given table<table border = "1" cellpadding = "5"><tr><td>X</td><td>70</td><td>80</td><td>90</td></tr><tr><td>P(X)</td><td>0.3</td><td>0.45</td><td>0.25</td></tr></table>Let's define a function C(X) such that it computes the cumulative probability up to and including that X value.


We'll define C(X) like so
C(70) = P(70) = 0.3
C(80) = P(70)+P(80) = 0.3+0.45 = 0.75
C(90) = P(70)+P(80)+P(90) = 0.3+0.45+0.25 = 1
So again, we define {{{C(X) = P(X <= k)}}} where k is a value in the domain of X.


So we have<table border = "1" cellpadding = "5"><tr><td>X</td><td>70</td><td>80</td><td>90</td></tr><tr><td>P(X)</td><td>0.3</td><td>0.45</td><td>0.25</td></tr><tr><td>C(X)</td><td>0.3</td><td>0.75</td><td>1</td></tr></table>



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Part (c)


Given table<table border = "1" cellpadding = "5"><tr><td>X</td><td>70</td><td>80</td><td>90</td></tr><tr><td>P(X)</td><td>0.3</td><td>0.45</td><td>0.25</td></tr></table>We see that the probability of getting 80 or higher, based on that table above, is 0.45+0.25 = 0.70
Simply add the probability values for X = 80 or larger.
There's a 70% chance of this happening.


Or, we could note that C(70) = 0.30
From this, we could then say 
P(𝑋 ≥ 80) = 1 - C(70)
P(𝑋 ≥ 80) = 1 - 0.30
P(𝑋 ≥ 80) = 0.70


This works because
P(70) + P(80) + P(90) = 1
The "P(80) + P(90)" portion is what we want while C(70) represents P(70)


In other words,
P(70) + P(80) + P(90) = 1
P(70) + [ P(80) + P(90) ] = 1
C(70) + P(𝑋 ≥ 80) = 1
P(𝑋 ≥ 80) = 1 - C(70)
P(𝑋 ≥ 80) = 1 - 0.3
P(𝑋 ≥ 80) = 0.7


Answer: 0.70
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