Respuesta :
Answer:
(A), (B), (E)
(A) The total area under the curve is 1.
True, by definition if we have a probability distribution we need to satisfy:
[tex] \int_{-\infty}^{\infty} f(x) dx =1[/tex]
Assuming off course that [tex] f(x) \geq 0 , \forall x[/tex]
(B) The proportion of data values between two numbers a and b is the area under the curve between a and b .
True. If we assume continuous random variable the probability is given by:
[tex] P(a \leq X \leq b) = \int_{a}^b f(x) dx = F(a) -F(b)[/tex]
(E) The curve is on or above the horizontal axis.
True. Is a conditions neccesary in order to satisfy a probability density function:
[tex] f(x) \geq 0, \forall x[/tex]
Because we can't have negative probabilities
Step-by-step explanation:
Let X a continuous random variable with probability density function [tex] f(x)[/tex]
(A) The total area under the curve is 1.
True, by definition if we have a probability distribution we need to satisfy:
[tex] \int_{-\infty}^{\infty} f(x) dx =1[/tex]
Assuming off course that [tex] f(x) \geq 0 , \forall x[/tex]
(B) The proportion of data values between two numbers a and b is the area under the curve between a and b .
True. If we assume continuous random variable the probability is given by:
[tex] P(a \leq X \leq b) = \int_{a}^b f(x) dx = F(a) -F(b)[/tex]
(C) The curve is symmetric and single-peaked.
False. Counterexample: The Chi square distribution is not symmetric but satisfy the conditions for a density function. Other counter examples are the Gamma or the Beta distribution.
(D) The curve satisfies the 68-95-99.7% rule.
False. This rule only applies for a normal distribution and not for all the other possible denisty functions like : Gamma, Beta, Exponential, Dirichlet, etc.
(E) The curve is on or above the horizontal axis.
True. Is a conditions neccesary in order to satisfy a probability density function:
[tex] f(x) \geq 0, \forall x[/tex]
Because we can't have negative probabilities
(A), (B), (E)
