Let X1,... Xn be independent random variables, all having the same distribution with expected value u and variance o?. The random variable X, defined as the arithmetic average of these variables, is called the sample mean. That is, the sample mean is given by n (a) Show that E[X] = µ. (b) Show that Var[X] = o²/n. The random variable S2, defined by E (Xi – X)² n – 1 is the sample variance. (Denominator is n – 1, not n, due to (d).) (c) Show that (Xi – X)? = E-, X? – nx.
Let X1,... Xn be independent random variables, all having the same distribution with expected value u and variance o?. The random variable X, defined as the arithmetic average of these variables, is called the sample mean. That is, the sample mean is given by n (a) Show that E[X] = µ. (b) Show that Var[X] = o²/n. The random variable S2, defined by E (Xi – X)² n – 1 is the sample variance. (Denominator is n – 1, not n, due to (d).) (c) Show that (Xi – X)? = E-, X? – nx.
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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![Let X1,... Xn be independent random variables, all having
the same distribution with expected value u and variance o?. The random
variable X, defined as the arithmetic average of these variables, is called
the sample mean. That is, the sample mean is given by
(a) Show that E[X] = µ.
(b) Show that Var[X] = o²/n.
The random variable S2, defined by
EL (Xi – X)²
п — 1
is the sample variance. (Denominator is n – 1, not n, due to (d).)
(c) Show that (Xi – X)? = E-, X? – nX.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F976365cc-1fb0-4972-89c5-069de40f599b%2Fd8f7f356-7c09-440a-9413-ec3e2ea5915a%2Famj4ybn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let X1,... Xn be independent random variables, all having
the same distribution with expected value u and variance o?. The random
variable X, defined as the arithmetic average of these variables, is called
the sample mean. That is, the sample mean is given by
(a) Show that E[X] = µ.
(b) Show that Var[X] = o²/n.
The random variable S2, defined by
EL (Xi – X)²
п — 1
is the sample variance. (Denominator is n – 1, not n, due to (d).)
(c) Show that (Xi – X)? = E-, X? – nX.
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