# Please help with density function question solving for sigma(x)?

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Homwork says:
find E(X), Var(X), sigma(X) of a random variable X
let x be given by its density function F(x) such that
f(x) = 0, if x <= 1
f(x) = 1/4, if 1 < x <= 5 ...show more

Best Answer

I'm assuming x is continuous.

E[x] = ∫xf(x)dx

E[x²] = ∫x²f(x)dx

Your density is just the constant 1/4 from 1 to 5. That makes this the uniform distribution from 1 to 5.

First verify that this is a pdf - the integral of f(x) from 1 to 5 must be 1

5 ........... 5

∫f(x)dx = ∫dx/4 = 1/4(5 - 1) = 1

1 .......... 1

So this is a pdf.

......... 5

E[x] = ∫xf(x)dx

......... 1

.......... 5

E[x] = ∫x/4dx

......... 1

E[x] = (1/4){(1/2)[5² - 1²]} = 24/8 = 3

For a uniform distribution from a to b the expected value is (a+b)/2 and in this case (1+5)/2=3

Var[x] = E[x²] - (E[x])²

.......... 5

E[x²] = ∫x²f(x)dx

.......... 1

.......... 5

E[x²] = ∫x²/4dx

.......... 1

E[x²] = (1/4){(1/3)[5³ - 1³]} = 124/12 = 31/3

Var[x] = E[x²] - (E[x])² = 31/3 - 3² = 4/3

The variance of the uniform distribution from a to b is just

(b-a)²/12 and in this case (5-1)²/12 = 16/12 = 4/3.

How did you get your answers? You might want to check where you made a mistake or if you think your answer is right - point out mine. :)

For sigma(x) I assume you mean the standard deviation which is just the square root of the variance.

E[x] = ∫xf(x)dx

E[x²] = ∫x²f(x)dx

Your density is just the constant 1/4 from 1 to 5. That makes this the uniform distribution from 1 to 5.

First verify that this is a pdf - the integral of f(x) from 1 to 5 must be 1

5 ........... 5

∫f(x)dx = ∫dx/4 = 1/4(5 - 1) = 1

1 .......... 1

So this is a pdf.

......... 5

E[x] = ∫xf(x)dx

......... 1

.......... 5

E[x] = ∫x/4dx

......... 1

E[x] = (1/4){(1/2)[5² - 1²]} = 24/8 = 3

For a uniform distribution from a to b the expected value is (a+b)/2 and in this case (1+5)/2=3

Var[x] = E[x²] - (E[x])²

.......... 5

E[x²] = ∫x²f(x)dx

.......... 1

.......... 5

E[x²] = ∫x²/4dx

.......... 1

E[x²] = (1/4){(1/3)[5³ - 1³]} = 124/12 = 31/3

Var[x] = E[x²] - (E[x])² = 31/3 - 3² = 4/3

The variance of the uniform distribution from a to b is just

(b-a)²/12 and in this case (5-1)²/12 = 16/12 = 4/3.

How did you get your answers? You might want to check where you made a mistake or if you think your answer is right - point out mine. :)

For sigma(x) I assume you mean the standard deviation which is just the square root of the variance.