Question

In: Statistics and Probability

1. Let X1 and X2 be continuous distributions uniformly distributed across the region [0, 2] ×...

1. Let X1 and X2 be continuous distributions uniformly distributed across the region [0, 2] × [0, 2] U (2, 4] × (2, 4].

a) Find the joint probability density function as well as marginal probability density function of X1 and X2.

b) Prove that X1 and X2 are not independent by showing cov(X1, X2) != 0.

Expert Solution

Given that X₁ and X₂ be two uniformly distributed random variables.

Regions are given to be:

(i.e) [ interval for x1*interval for x2] U [interval for x1*interval for x2]

Since union of cartesian products of intervals are given as regions we split the intervals for X₁ and X₂.

(i.e) X₁ [0,2]U(2,4] and X₂ [0,2]U(2,4]

The probability density function of uniform distribution is given by,

where a<x<b, a and b are constants.

(a)JOINT PROBABILITY DENSITY FUNCTION OF X₁ AND X₂:

MARGINAL DENSITY FUNCTION OF X1 :

MARGINAL DENSITY FUNCTION OF X2 :

(b)COVARIANCE OF X1 AND X2:

The formula for covariance is

Cov(X1,X2)=E(X1*X2)-E(X1)*E(X2)

Thus

THUS X1 AND X2 ARE NOT INDEPENDENT.

orchestra answered 2 years ago

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