Some more stats trouble -

**bill2010** · January 7th 2009, 09:26 AM

My books and notes fail me

, i would greatly appreciate help and explanation of the following problems.

a)
The PDF of a continuous random variable, (X) is shown below:

f(x) = ke-2x, x > 0,
f(x) = 0, otherwise.

(i)Calculate the value of the constant, k.
(ii)Find P(1 £ X £ 2).

b)
The PDF of a continuous random variable, (X), is shown below:

f(x) = cx-1/2, 0 < x < 9,
f(x) = 0, otherwise.

(i)Calculate the value of the constant, c.
(ii)Find the expected value of X.
(iii)Find the value of the variance of X.

Thanks!

**mr fantastic** · January 7th 2009, 02:27 PM

Originally Posted by bill2010

My books and notes fail me

, i would greatly appreciate help and explanation of the following problems.

a)
The PDF of a continuous random variable, (X) is shown below:

f(x) = ke-2x, x > 0,
f(x) = 0, otherwise.

(i)Calculate the value of the constant, k.
(ii)Find P(1 £ X £ 2).

b)
The PDF of a continuous random variable, (X), is shown below:

f(x) = cx-1/2, 0 < x < 9,
f(x) = 0, otherwise.

(i)Calculate the value of the constant, c.
(ii)Find the expected value of X.
(iii)Find the value of the variance of X.

Thanks!

a) (i) Solve $\int_0^{\infty} f(x) \, dx = 1$ for k.

(ii) Calculate $\int_1^{2} f(x) \, dx$ (substitute your value of k from (i)).

b) (i) Solve $\int_0^{9} f(x) \, dx = 1$ for c.

(ii) Calculate $E(X) = \int_0^{9} x \, f(x) \, dx$ (substitute your value of c from (i)).

(iii) Calculate $E(X^2) = \int_0^{9} x^2 \, f(x) \, dx$ (substitute your value of c from (i)). Then $Var(X) = E(X^2) - [E(X)]^2$ .

**bill2010** · January 7th 2009, 03:32 PM

Thankyou very much for what you have done, i am starting to understand it much better now.

I am however doing integration for the 1st time, and im not sure how to correctly solve and display my workings from start to finish.
Could you explain the proper method?

Thanks again!

**mr fantastic** · January 8th 2009, 09:25 PM

Originally Posted by bill2010

Thankyou very much for what you have done, i am starting to understand it much better now.

I am however doing integration for the 1st time, and im not sure how to correctly solve and display my workings from start to finish.
Could you explain the proper method?

Thanks again!

Have you attempted the integrations? Please show what you've done.

**bill2010** · January 8th 2009, 09:58 PM

I havent got any workings, i dont know where to start.

I have notes covering examples of CRV's but nothing in my notes, slides or books explains how to deal with constants, especially 2 of them and a negative exponent...

(the correct notation - copy and paste ruined the above)

f(x) = ke-2x

and

f(x) = cx-1/2

I understand a little that integration is like a reverse of differentiation, my maths skills are pretty basic and this stuff can be hard to crack the 1st time round. If you could explain what the formulas you added mean that would be so much help, thanks!

**mr fantastic** · January 8th 2009, 10:04 PM

Originally Posted by bill2010

I havent got any workings, i dont know where to start.

I have notes covering examples of CRV's but nothing in my notes, slides or books explains how to deal with constants, especially 2 of them and a negative exponent...

(the correct notation - copy and paste ruined the above)

f(x) = ke-2x

and

f(x) = cx-1/2

I understand a little that integration is like a reverse of differentiation, my maths skills are pretty basic and this stuff can be hard to crack the 1st time round. If you could explain what the formulas you added mean that would be so much help, thanks!

If you don't know how to integrate there's no point me posting the solution. You need to take one ot two steps back and get integration under control. Only then will you be able to move forward with stats questions like this one.

**bill2010** · January 8th 2009, 10:15 PM

I have examples of integration to go by, but it deals specifically with a (0< x >6) example. Getting information about this topic has been near impossible, ive got 5 books barely containing anything on how to use integration with CVR's.

**mr fantastic** · January 9th 2009, 01:21 AM

Originally Posted by bill2010

I have examples of integration to go by, but it deals specifically with a (0< x >6) example. Getting information about this topic has been near impossible, ive got 5 books barely containing anything on how to use integration with CVR's.

Originally Posted by mr fantastic

a) (i) Solve $\int_0^{\infty} f(x) \, dx = 1$ for k.

(ii) Calculate $\int_1^{2} f(x) \, dx$ (substitute your value of k from (i)).

b) (i) Solve $\int_0^{9} f(x) \, dx = 1$ for c.

(ii) Calculate $E(X) = \int_0^{9} x \, f(x) \, dx$ (substitute your value of c from (i)).

(iii) Calculate $E(X^2) = \int_0^{9} x^2 \, f(x) \, dx$ (substitute your value of c from (i)). Then $Var(X) = E(X^2) - [E(X)]^2$ .

(a) (i) $\int_0^{\infty} f(x) \, dx = 1$ :

$\int_0^{\infty} f(x) \, dx = \int_0^{\infty} k e^{-2x} \, dx = \left[- \frac{k}{2} e^{-2x}\right]^{+\infty}_{x=0} = 0 - (-\frac{k}{2}) = \frac{k}{2}$ .

Therfore $\frac{k}{2} = 1 \Rightarrow k = 2$ .

(b) (i) $\int_0^{9} f(x) \, dx = 1$ :

$\int_0^{9} c x^{-1/2} \, dx = \left[ 2c x^{1/2}\right]_{0}^{9} = 6c - 0 = 6c$ .

Therefore $6c = 1 \Rightarrow c = \frac{1}{6}$ .

**bill2010** · January 9th 2009, 10:48 AM

Thanks so much!

Sorry ive been a pain.

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