Question

In: Statistics and Probability

Let A be a non-negative random variable (A>0) a) If A is discrete, show that E[A]...

Let A be a non-negative random variable (A>0)

a) If A is discrete, show that E[A] >= 0

b) If A is continous , show E[X]>=0

Expert Solution

a)

A discrete random variable A has a countable number of possible values.

Example: Two dice are thrown. Let A represent the sum of two dice. In this case, A is non negtative.

Then the probability distribution of X is as follows:

A	2	3	4	5	6	7	8	9	10	11	12
P (A)	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36

E(A) = ∑ A∙ P(A)

= [ 2∙(1/36) ] + [ 3∙(2/36) ] + [ 4∙(3/36) ] + [ 5∙(4/36) ] + [ 6∙(5/36) ] + [ 7∙(6/36) ] + [ 8∙(5/36) ] +

[ 9∙(4/36) ] + [ 10∙(3/36) ] + [ 11∙(2/36) ] + [ 12∙(1/36) ]

= [ 2 + 6 + 12 + 20 + 30 + 42 + 40 + 36 + 30 + 22 + 12 ] / 36

= 252 / 36

E(A) = 7

b)

Let A = Continuous random variable and it is defined as

A = x²; 0 < x < 3

= 0 ; Otherwise

E[x] =

=

= 1/4 { ( x⁴ )_{3 -} ( x⁴ )₀}

= 1 / 4 [ 3⁴]

E[X] = 81 / 4

E[X] = 20.25

In both the cases Expected value is greater than zero.

orchestra answered 2 years ago

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