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Apprentice
Karma Points: 116
Respect (100%):
Date Posted: 6/17/2008 8:50:15 PM  Status: Closed
A question of probability
Course Textbook Chapter Problem
Statistics and Probability N/A N/A N/A
Question Details:
Dear everyone,
 
I have a question to be answered, I will rate high if the answer is satisfied!
 
Let X equal the number of flips of a fair coin that are required to observe the same face on consecutive flips.
(a) Find the p.m.f of X.    (b) Give the values of the mean, variance, and standard
deviation of X.   (c) Find the values of P(X<=3),  P(X>=5) ,  P(X=3)
 
For the question (a), the answer is  f(X) = (x-1) / (2^x), I don't know how to get this pmf, hopefully someone could give me the explanation.  Thanks. 

Answers:

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Oracle
Karma Points: 12,589
(University of Moncton, Canada)
Date Posted: 6/20/2008 6:42:53 AM  Status: Live
Asker's Rating: Lifesaver   
Response:
(a) Before finding the pmf of f, first you'll need to find the cumulative function of X noted F(x)
 
Let X equal the number of flips of a fair coin that are required to observe the same face on consecutive flips.
 
Let's say the first coin was heads, then all the coins after that needs to land on heads so the probability becomes
 
 
Each time you have 50% chance of having heads and its independant events so you multiply all the events which are all 50%.
 
 
Throwing a coin is a discrete variable so this is why we use sum's instead of integrals.
 
The pmf of X is noted f(x) and thats the sum of F(x).
 
 
 
This is a geometric serie that converges to
That's how you come up with
 
(b) Mean of X =
 
Variance of X =
 
We found E(x) already above... but to find E(x2) its pretty much the same thing
 
 
Standard deviation of X =
 
(c) P(x<=3) = P(x=1) + P(x=2) + P(x=3) = 1/2 + 1/4 + 1/8 = 7/8
P(x>=5) = 1-P(x<4) = 1-P(x<=3) = 1-7/8 = 1/8
P(x=3) = 1/8
APPOLLO's Comment:
Perfect explanations, thank you!!
Please rate me for the time it takes to help you!!! ty.
 
Did you know if you mesure the length of your arm and divide by the length of your elbow to your finger you will get
As of today in 2008, nobody in the world has found the sum of all they know is that it converges because of the p-serie theorem but they cannot find the sum.

One of my teachers, as been working on that sum for about 5 years now, he found a concret answer (he ended up creating an instrument of that serie called ''tritare'' its pretty cool) but he has not found a mathematical expression to express it.
 
For more info:
 
 
My pic is the instrument called Tritare.
 



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