Answered: Question 2 • Find the probability P(x ≤… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.4: Values Of The Trigonometric Functions

Problem 24E

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Question

please answer question 2

Another application of integration is in probability. This task applies your algebra
and calculus skills to this domain.
If X is a continuous random variable with range [a, b] then the probability density
function satisfies the following two properties:
Property 1: f(x) ≥ 0 for all x = [a, b].
Property 2:
TASK 1: INTEGRATION
b
[² f
f(x) dx = 1.
Question 1
(1) Explain why the following functions are, or are not, probability density functions:
(a) f(x) = 2 for x = [1, √e]
(b) f(x) = 5 (9x² — xª) for x = [0, 3]

Given a probability density function f(x) with range [a, b], we can obtain the
probability that X is at most k
from the area of a region under the function.
P(X ≤ k) = f* f(x)dx
a
To find the probability that c≤ x ≤k:
rk
P(c ≤ x ≤ k) = √ k f(x)dx
Note: P(X <k) = P(X ≤ k).
Question 2
• Find the probability P(x ≤ 3) given the probability density function you identi-
fied in Question 1.
Give an exact answer, and an approximate answer to two decimal places.
• Why is Property 1 in Question (1) essential for a function to be a probability
density function?

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