Question

The arithmetic mean of a certain distribution is 5. The second and the third moments

about mean are 20 and 140 respectively. Find the third moment of the distribution

about 10.​

Answer 1

Answer:

I hope you have got your answer see it

Answer 2

Answer:

he third moment of the distribution of about 10 is -285.

Step-by-step explanation:

Call the random variable x.

Now, define a new variable y = x - 5. Note that x - 10 = y - 5.

So, it is clear that (x-10)³ = (y-5)³

Also, note that (y-5)³ can be expanded as follows:

y³ - 15y² + 75y - 125

Letting E denote expectation with respect to the random variable x, we see that

E[(y-5)³] = E(y³) -15 E(y²) + 75 E(y) - 125.. (1)

Again, recalling that y = x - 5, we have

E(y^3) = 140

E(y^2) = 20

E(y) = E(x) - 5 = 5 - 5 = 0

Thus, substituting in equation (1):

E[(y-5)³] = 140 -15(20) + 75(0) -125 = -285

But as y = x - 5 we get that:

E[(y-5)³] = E[({x-5} -5)³] = E[(x-10)³]

So, E[(x-10)³] = -285.

Therefore, the third moment of the distribution of about 10 is -285.

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