Find the mean and the variance of the random variable X with probability function or
Chapter 24, Problem 24.1.99(choose chapter or problem)
TEAM PROJECT. Means, Variances, Expectations.
(a) Show that E(X - \(\mu\)) = 0, \(\sigma^{2}=E\left(X^{2}\right)-\mu^{2}\).
(b) Prove (10)-(12).
(c) Find all the moments of the uniform distribution on an interval \(a \leqq x \leqq b\).
(d) The skewness 'Y of a random variable X is defined by
(13) \(\gamma=\frac{1}{\sigma^{3}} E\left([X-\mu]^{3}\right)\).
Show that for a symmetric distribution (whose third central moment exists) the skewness is zero.
(e) Find the skewness of the distribution with density \(f(x)=x e^{-x}\) when x > 0 and fIx) = 0 otherwise. Sketch f(x).
(I) Calculate the skewness of a few simple discrete distributions of your own choice.
(g) Find a non symmetric discrete distribution with 3 possible values. mean O. and skewness O.
Text Transcription:
mu
sigma^2 = E(X^2) - mu^2
a <= x <= b
gamma = 1 / sigma^3 E([X - mu]^3)
f(x) = xe^-x
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