# Some more stats trouble -

• Jan 7th 2009, 09:26 AM
bill2010
Some more stats trouble -
My books and notes fail me (Worried), 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!
• Jan 7th 2009, 02:27 PM
mr fantastic
Quote:

Originally Posted by bill2010
My books and notes fail me (Worried), 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 $\displaystyle \int_0^{\infty} f(x) \, dx = 1$ for k.

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

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

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

(iii) Calculate $\displaystyle E(X^2) = \int_0^{9} x^2 \, f(x) \, dx$ (substitute your value of c from (i)). Then $\displaystyle Var(X) = E(X^2) - [E(X)]^2$.
• Jan 7th 2009, 03:32 PM
bill2010
Thankyou very much for what you have done, i am starting to understand it much better now.(Clapping)

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!
• Jan 8th 2009, 09:25 PM
mr fantastic
Quote:

Originally Posted by bill2010
Thankyou very much for what you have done, i am starting to understand it much better now.(Clapping)

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.
• Jan 8th 2009, 09:58 PM
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!
• Jan 8th 2009, 10:04 PM
mr fantastic
Quote:

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.
• Jan 8th 2009, 10:15 PM
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.(Crying)
• Jan 9th 2009, 01:21 AM
mr fantastic
Quote:

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.(Crying)

Quote:

Originally Posted by mr fantastic
a) (i) Solve $\displaystyle \int_0^{\infty} f(x) \, dx = 1$ for k.

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

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

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

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

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

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

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

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

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

Therefore $\displaystyle 6c = 1 \Rightarrow c = \frac{1}{6}$.
• Jan 9th 2009, 10:48 AM
bill2010
Thanks so much!

Sorry ive been a pain.