Question 1192780: Determine the cumulative distribution function for the random variable.
X -2 -1 0 1 2
f(x) 1/8 2/8 2/8 2/8 1/8
(a)P(X≤2)
(b)P(X>-2)
(c)P(-1≤X≤1)
(d)P(X≤-1 or X=2)
also determine the following probabilities:
(a)P(X≤1.15)
(b)P(X≤2.2)
(c)P(-1.1
(d)P(X>0)
Answer by math_tutor2020(3816)   (Show Source):
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Given Table
| X | -2 | -1 | 0 | 1 | 2 | | P(X) | 1/8 | 2/8 | 2/8 | 2/8 | 1/8 |
The notation P(X ≤ 2) means "The probability X ≤ 2 occurs"
This means X = 2 or X < 2.
As the table shows, the largest item is X = 2 itself, meaning that X ≤ 2 encompasses all possible outcomes. Therefore, we have 100% probability that X ≤ 2 happens.
We write P(X ≤ 2) = 1
Note that all the fractions add to: 1/8+2/8+2/8+2/8+1/8 = (1+2+2+2+1)/8 = 8/8 = 1.
P(X > -2) = 7/8 because P(X = -2) = 1/8, so being larger than -2 will have a probability of 1 - 1/8 = 7/8.
Or you could add up all the P(x) values for X = -1 all the way up to X = 2.
P(-1 ≤ X ≤ 1) = P(-1) + P(0) + P(1)
P(-1 ≤ X ≤ 1) = 2/8 + 2/8 + 2/8
P(-1 ≤ X ≤ 1) = (2+2+2)/8
P(-1 ≤ X ≤ 1) = 6/8
P(-1 ≤ X ≤ 1) = 3/4
P(X ≤ -1) = P(-2) + P(-1)
P(X ≤ -1) = 1/8 + 2/8
P(X ≤ -1) = 3/8
P(X ≤ -1 or X = 2) = P(X ≤ -1) + P(X = 2)
P(X ≤ -1 or X = 2) = 3/8 + 1/8
P(X ≤ -1 or X = 2) = 4/8
P(X ≤ -1 or X = 2) = 1/2
This is a discrete probability distribution with X only able to take on the values from the set {-2, -1, 0, 1, 2}
Something like X = 1.15 is not possible.
Saying X ≤ 1.15 is the same as X ≤ 1 because X = 1 satisfies the requirements that X = 1.15 or X < 1.15.
So P(X ≤ 1.15) = P(X ≤ 1) = 7/8 which can be found by adding every P(X) value from x = -2 on up to x = 1; or take the shortcut 1 - P(2) = 1 - 1/8 = 7/8.
P(X ≤ 2.2) = P(X ≤ 2) = 1 for similar reasoning as mentioned earlier.
P(-1.1 < X ≤ 1) = P(-1 < X ≤ 1)
P(-1.1 < X ≤ 1) = 3/4 also found earlier
P(X > 0) = P(X ≥ 1)
P(X > 0) = P(1) + P(2)
P(X > 0) = 2/8 + 1/8
P(X > 0) = 3/8
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Answers:
Part 1
(a)P(X ≤ 2) = 1
(b)P(X > -2) = 7/8
(c)P(-1 ≤ X ≤ 1) = 3/4
(d)P(X ≤ -1 or X=2) = 1/2
Part 2
(a)P(X ≤ 1.15) = 7/8
(b)P(X ≤ 2.2) = 1
(c)P(-1.1 < X ≤ 1) = 3/4
(d)P(X > 0) = 3/8
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