Asked by Chloe Avenaim on Sep 27, 2024
Verified
Suppose f(x) = 1/4 over the range a ≤ x ≤ b,and suppose P(X > 4) = 1/2.What are the values for a and b?
A) 0 and 4
B) 2 and 6
C) Can be any range of x values whose length (b − a) equals 4.
D) Cannot answer with the information given.
Probability Density
A mathematical function that specifies the likelihood of a continuous variable taking on a particular value, critical in the context of continuous probability distributions.
Uniform Distribution
A type of probability distribution where all outcomes are equally likely to occur within a certain range.
- Calculate probabilities for continuous uniform distributions.
Verified Answer
TM
tebogo magopane1 day ago
Final Answer :
B
Explanation :
Since P(X > 4) = 1/2, this means that half of the total area under the curve lies to the right of x = 4. Since the given function f(x) is constant over the range a ≤ x ≤ b, the total area under the curve can be written as:
total area = f(x) * (b - a) = 1/4 * (b - a)
Half of this area lies to the right of x = 4, so we can write:
1/2 = (1/4) * (b - 4)
Solving for b, we get:
b = 2 + 4 = 6
Similarly, half of the total area lies to the left of x = 2. So we can write:
1/2 = (1/4) * (2 - a)
Solving for a, we get:
a = 2 - 2 = 0
Therefore, the values for a and b are a=0 and b=6, which corresponds to choice B.
total area = f(x) * (b - a) = 1/4 * (b - a)
Half of this area lies to the right of x = 4, so we can write:
1/2 = (1/4) * (b - 4)
Solving for b, we get:
b = 2 + 4 = 6
Similarly, half of the total area lies to the left of x = 2. So we can write:
1/2 = (1/4) * (2 - a)
Solving for a, we get:
a = 2 - 2 = 0
Therefore, the values for a and b are a=0 and b=6, which corresponds to choice B.
Learning Objectives
- Calculate probabilities for continuous uniform distributions.
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