- x)² – 6 0< x < 2 otherwise, psitive. the cdf of X. nt Ber(0) random variables and le nerating function of X1. cenerating function of Y. ctation and variance of Y. the distribution of Y. nat the fuel efficiency of its vehicle of 47 miles per gallon (mpg). T ciencies of a simple random samp. e mean of the 15 measurements wa
- x)² – 6 0< x < 2 otherwise, psitive. the cdf of X. nt Ber(0) random variables and le nerating function of X1. cenerating function of Y. ctation and variance of Y. the distribution of Y. nat the fuel efficiency of its vehicle of 47 miles per gallon (mpg). T ciencies of a simple random samp. e mean of the 15 measurements wa
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
Part C
![1. (a) Let X be a continuous random variable with pdf
Sc(1 – x)? – b 0 <¤< 2
fx(x) =
otherwise,
for constants c>0 and b.
(i) Show that b cannot be positive.
(ii) Show that c =
(1+ 26).
Let b = -:
(iii) Find Fx(1), where Fx is the cdf of X.
(iv) Find P(X = 0).
(v) Find E[X].
(b) Let X1,..., Xn be independent Ber(0) random variables and let
Y = X1 +
...+ Xn.
(i) Calculate the moment generating function of X1.
(ii) Hence find the moment generating function of Y.
(iii) Hence calculate the expectation and variance of Y.
(iv) Write down the name of the distribution of Y.
(c) A car manufacturer claims that the fuel efficiency of its vehicles
meets a regulatory standard of 47 miles per gallon (mpg). To
assess this claim, the fuel efficiencies of a simple random sample
of 15 cars were measured. The mean of the 15 measurements was
= 46.75 mpg and their standard deviation was s 0.72 mpg.
y =
Assume that the measurements follow a Normal distribution.
(i) Conduct a test of size 5% of the null hypothesis that the
population mean efficiency is 47 mpg against the alternative
hypothesis that the population mean efficiency is less than 47
mpg. State clearly your test statistic, its null distribution, the
critical region, and your conclusion about the manufacturer's
claim.
(ii) Compute an equal-tailed 95% confidence interval for the
population mean efficiency.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73b88bb0-7d64-4a0b-8462-ec4405bd1dbb%2F7606fb2d-5c5b-4b72-9d98-168906c6894f%2F0lrmg4f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. (a) Let X be a continuous random variable with pdf
Sc(1 – x)? – b 0 <¤< 2
fx(x) =
otherwise,
for constants c>0 and b.
(i) Show that b cannot be positive.
(ii) Show that c =
(1+ 26).
Let b = -:
(iii) Find Fx(1), where Fx is the cdf of X.
(iv) Find P(X = 0).
(v) Find E[X].
(b) Let X1,..., Xn be independent Ber(0) random variables and let
Y = X1 +
...+ Xn.
(i) Calculate the moment generating function of X1.
(ii) Hence find the moment generating function of Y.
(iii) Hence calculate the expectation and variance of Y.
(iv) Write down the name of the distribution of Y.
(c) A car manufacturer claims that the fuel efficiency of its vehicles
meets a regulatory standard of 47 miles per gallon (mpg). To
assess this claim, the fuel efficiencies of a simple random sample
of 15 cars were measured. The mean of the 15 measurements was
= 46.75 mpg and their standard deviation was s 0.72 mpg.
y =
Assume that the measurements follow a Normal distribution.
(i) Conduct a test of size 5% of the null hypothesis that the
population mean efficiency is 47 mpg against the alternative
hypothesis that the population mean efficiency is less than 47
mpg. State clearly your test statistic, its null distribution, the
critical region, and your conclusion about the manufacturer's
claim.
(ii) Compute an equal-tailed 95% confidence interval for the
population mean efficiency.
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