please answer question 2 from the second image using question 1 Given a probability density function f(x) with range [a, b], we can obtain theprobability that X is at most kfrom the area of a region under the function.P(X ≤ k) = f* f(x)dxaTo find the probability that c≤ x ≤k:rkP(c ≤ x ≤ k) = √ k f(x)dxNote: P(X Another application of integration is in probability. This task applies your algebraand calculus skills to this domain.If X is a continuous random variable with range [a, b] then the probability densityfunction satisfies the following two properties:Property 1: f(x) ≥ 0 for all x = [a, b].Property 2:TASK 1: INTEGRATIONb[² ff(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]

a)Given, f(x)=2x x∈[1,e]Here f(x)≥0 ,for all x∈[1,e]Now we will check Property 2P(1≤X≤e)=∫1e 2xdx…

Answered: Property 1: f(x) ≥ 0 for all x € [a,…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Related questions

Question

please answer question 2 from the second image using question 1

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?

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]

Expert Solution

Step 1

a)Given,

$f (x) = \frac{2}{x} x \in [1, \sqrt{e}] H e r e f (x) \geq 0, f o r a l l x \in [1, \sqrt{e}] N o w w e w i l l c h e c k P r o p e r t y 2 P (1 \leq X \leq \sqrt{e}) = \int_{1}^{\sqrt{e}} \frac{2}{x} d x = 2 {[\ln (x)]}_{1}^{\sqrt{e}} = 1 s o f (x) i s p d f N o w f r o m q u s t i o n 2 P (X \leq \frac{3}{2}) = \int_{1}^{\frac{3}{2}} \frac{2}{x} d x = 2 {[\ln (x)]}_{1}^{\frac{3}{2}} = 2 [\ln 3 - \ln 2] [a s, \ln (1) = 0)$