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.
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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![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%2Fff49b279-fc11-439e-9d67-d425940742a5%2Falthmu_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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