From Kaplan:
Example: Discrete uniform distribution
Determine p(6), F(6), and P(2 ≤ X ≤ 8) for the discrete uniform distribution function defined as:
X = {2, 4, 6, 8, 10}, p(x) = 0.2
Answer:
p(6) = 0.2, since p(x) = 0.2 for all x. F(6) = P(X ≤ 6) = np(x) = 3(0.2) = 0.6. Note that n = 3 since 6 is the third outcome in the range of possible outcomes.
What does F(6) stand for & how were the particulars used to find F(6) = 0.6 ?
F is called a cumulative distribution function: F(x)=P(X ≤ x) where X is a random variable and x is a specified constant.
In this case, F(6) means the probability of getting a value less than or equal to 6. This includes the values 2, 4, and 6. Since each value has a uniform probability of 0.2 of occurring, then F(6) = P(X≤ 6) = p(2) + p(4) + p(6) = 0.2 + 0.2 + 0.2 = 3 * 0.2 = 0.6
Perfect explanation, thank you!
You already wrote it:
F(6) = P(X ≤ 6) = P(X = 2) + P(X = 4) + P(X = 6)
In addition to Sartoral’s question, Kaplan also explained this:
“P(2 ≤ X ≤ 8) = 4(0.2) = 0.8. Note that k= 4, since there are four outcomes in the range 2 ≤ X ≤8”. What is wrong if i solve this by P(2 ≤ X ≤ 8) = F(8) - F(2) = 0.8 - 0.2 = 0.6, which is different to the answer. Can someone please explain to me, thank you <3
F(x) = P(X ≤ x).
So F(8) − F(2) = P(X ≤ 8) − P(X ≤ 2) = P(2 < X ≤ 8) ≠ P(2 ≤ X ≤ 8).
Note that the difference is P(X = 2), which is exactly how much your answer differs from the correct answer.
Thank you so much sir!
My pleasure.