Extremely Tedious Math/Statistics Homework Question

[number0]

Homework Statement

A coin with probability p of heads is tossed until the first head occurs. It is then tossed again
until the first tail occurs. Let X be the total number of tosses required.

(a) Find the PMF of X.
(b) Find the mean and variance of X.

Homework Equations

PMF is basically a function that helps calculate all the probability of all the possible values x can take, P(X = x).

By "mean", I think my professor means the expected value of X. Here is a link to what expected value for discrete values is:

http://en.wikipedia.org/wiki/Expected_value#Discrete_random_variable.2C_finite_caseThe variance of X is given by the following formula listed in the link below:

http://en.wikipedia.org/wiki/Variance#Discrete_random_variable

The Attempt at a Solution

I apologize if I cannot post in latex format or any kind of format for that matter. Instead, I will upload my work onto paint.

Part a) was relatively straightforward. I plugged in values X = 0, X = 1 ... X = x and came up with a formula. For example, when X = 4 we have:

H H H T
T H H T
T T H T

The formula for Part a) is display in the file I uploaded. Part b) is the hard one. I know what formulas to use and how to set it up, but it seems like it is just too TEDIOUS to calculate. Letting Wolframalpha do the work for me, the expected value of X is 1/(p-p^2). I really want to understand how I am suppose to derive the E(X) and the Var(X) in a more simple manner. Furthermore, I am not so sure if the formula I used in Part a) could be much more simplified. Anyways, I would appreciate it if someone can help me on this problem.

Thank you for your time and response.

 

[Ray Vickson]

Homework Statement

A coin with probability p of heads is tossed until the first head occurs. It is then tossed again
until the first tail occurs. Let X be the total number of tosses required.

(a) Find the PMF of X.
(b) Find the mean and variance of X.

Homework Equations

PMF is basically a function that helps calculate all the probability of all the possible values x can take, P(X = x).

By "mean", I think my professor means the expected value of X. Here is a link to what expected value for discrete values is:

http://en.wikipedia.org/wiki/Expected_value#Discrete_random_variable.2C_finite_case


The variance of X is given by the following formula listed in the link below:

http://en.wikipedia.org/wiki/Variance#Discrete_random_variable

The Attempt at a Solution

I apologize if I cannot post in latex format or any kind of format for that matter. Instead, I will upload my work onto paint.

Part a) was relatively straightforward. I plugged in values X = 0, X = 1 ... X = x and came up with a formula. For example, when X = 4 we have:

H H H T
T H H T
T T H T

The formula for Part a) is display in the file I uploaded.


Part b) is the hard one. I know what formulas to use and how to set it up, but it seems like it is just too TEDIOUS to calculate. Letting Wolframalpha do the work for me, the expected value of X is 1/(p-p^2). I really want to understand how I am suppose to derive the E(X) and the Var(X) in a more simple manner. Furthermore, I am not so sure if the formula I used in Part a) could be much more simplified.


Anyways, I would appreciate it if someone can help me on this problem.

Thank you for your time and response.

Your statement "Part b) is the hard one" is exactly opposite to the truth. If X1 =number of tosses until the first H and then X2 = number of tosses until the first T, the total number is X = X1 + X2. Do you know how to find the mean and variance of a sum of random variables?

RGV

 

[number0]

Your statement "Part b) is the hard one" is exactly opposite to the truth. If X1 =number of tosses until the first H and then X2 = number of tosses until the first T, the total number is X = X1 + X2. Do you know how to find the mean and variance of a sum of random variables?

RGV

Hmmm... I never thought to break up the problem into two. It should be much easier now. Thanks.