# Probability: Random Variable, Mean, Variance & Density Function

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(1) The random variable X takes on the values 0, 1,2,3 with respective probabilities

1 12 48 64 125' 125' 125' 125

(a) Find E(X),E(X2) and var (X) (b) Find E((3X ± 2)2 (hint: easy if you use the results in (a))

(2) A contractor's profit on a construction job is a random variable with density function

1 f(x) = 18(x+ 1), —1 < x < 5 --= 0 otherwise

(a) Find the expected profit, the variance and the standard deviation (b) Determine the probability that the profit falls within two standard deviations of the mean and compare the result with Chebyshev's Theorem

(3) Find the moment generating function for the uniform random variable on [0,1] and use it to find the mean and the variance. Recall

f(x) = 1, 0 < x < 1

:=0 otherwise

(4) Explain why there can be no random variable whose moment generating function is m(t) = rt-t- . (hint: look at the potential variance and recall. that the variance must be positive

(5) The length of time for an individual to wait at a lunch counter is a random variable whose density function is

f(x) =1 -4 e-riz, x > 0

= 0 otherwise.

(a) Find the mean and variance (b) Find the probability that the random variable is within three standard deviations of the mean and compare with Chebyshev's Theorem.

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Statistics: Probability density function, central limit theorem

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If X1,X2,..., Xn, are (iid) , from a distribution with mean μ and variance σ^2. Define the sample mean as

Xbar = (X1+X2+...+Xn) / n

(a) Show that the mean and variances of the probability density function of Xbar are given as E(Xbar) = μ

Var(Xbar) = (σ^2)/n

b) What is the central limit theorem?

c) If n, is large, can you describe fully, the probability density function of Xbar?

d) Can you describe fully the probability density function of the variable y = e^Xbar? This random variable is called a lognormal random variable, and is used very frequently in finance.