GalwayMathsStudent
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- Sep 14, 2016
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Over the past few weeks i have been working my way through "John E. Freund's Mathematical Statistics with Application" and I have gotten stuck on a few problems over the first 4 sections. I would appreciate and tips/solutions towards any of the problems which i will post below:
(I will use aIntb to denote integral from a to b)
(Solved)Q 1.15: Repeatedly apply nCr=(n-1)Cr+(n-1)C(r-1) to show: nCr= sum from i=1 to r+1 ( (n-i)C(1-i+1) )
A: Was told its just induction proof.
(Solved)Q 4.36: Explain why there can be no random variable for which Mx(t)=t/(1-t) , (Mx(t) is the moment-generating function of the random variable x.)
A: I tried differentiating it twice and got mean of x=1 and variance of x=1 which seems fine. Maybe its because Mx(o) is not equal to 1?
It is because Mx(o) is not equal to 1. as E[x^0]=E[1]=1=Integral(Mx(0)f(x)dx
and for this integral to be one Mx(0)=1.
(Unsolved)Q 4.59: (a) Show that the conditional distribution function of the continuous random variable X,given by a<X<=b,given by
E[u(x)|a<x<=b]= (aIntb(u(x)f(x)dx)) / (aIntb( f(x)dx)
A: no idea for either part although I know d/dx(F(x))=f(x) is used for part b.
(Unsolved)Q 4.67: A and B are betting on a repeated flips of a coin. At the start of the game A has a dollars and B has b dollars, at each flip the loser pays the winner one dollar, and the game continues until either player is "ruined". Making use of the fact that in an equitable game each player's mathematical expectation is zero, find the probability that A will win B's b dollars before he loses his a dollars.
A: no idea with this one thought about the binomial formula but realized that would not work.(Answer is supposed to be a/(a+b)
(Solved)Q 4.83: The amount of time(in minutes) that an executive of a certain firm talks on the telephone is a random variable having the probability density:
A: The formula from the question is( E[u(x)|a<x<=b]= (aIntb(u(x)f(x)dx)) / (aIntb( f(x)dx))
I tried E[x|x>=1]= (1Int2(x*(x/4)dx)) / (1Int2((x/4)dx)) + (2Int(infinity)(x*(4/x^3)dx)) / (2Int(infinity)((4/x^3)dx))
=(14/6)/(9/6)+4=5.55555 minutes
but the back of the books says the answer is 2.95 mins so i don't know where i went wrong.
Essentially should be (j1+j2)/(j3+j4) not j1/j3+j2/j4
Any help would be appreciated.
(I will use aIntb to denote integral from a to b)
(Solved)Q 1.15: Repeatedly apply nCr=(n-1)Cr+(n-1)C(r-1) to show: nCr= sum from i=1 to r+1 ( (n-i)C(1-i+1) )
A: Was told its just induction proof.
(Solved)Q 4.36: Explain why there can be no random variable for which Mx(t)=t/(1-t) , (Mx(t) is the moment-generating function of the random variable x.)
A: I tried differentiating it twice and got mean of x=1 and variance of x=1 which seems fine. Maybe its because Mx(o) is not equal to 1?
It is because Mx(o) is not equal to 1. as E[x^0]=E[1]=1=Integral(Mx(0)f(x)dx
and for this integral to be one Mx(0)=1.
(Unsolved)Q 4.59: (a) Show that the conditional distribution function of the continuous random variable X,given by a<X<=b,given by
{ 0 for x<=a
F(x)={ (F(x)-F(a)) / (F(b)-F(a)) for a<x<=b{ 1 for x>b
(Solved part b)(b) differentiate (a) with respect to x to find the Conditional density of X given a<X<=b, and show thatE[u(x)|a<x<=b]= (aIntb(u(x)f(x)dx)) / (aIntb( f(x)dx)
A: no idea for either part although I know d/dx(F(x))=f(x) is used for part b.
(Unsolved)Q 4.67: A and B are betting on a repeated flips of a coin. At the start of the game A has a dollars and B has b dollars, at each flip the loser pays the winner one dollar, and the game continues until either player is "ruined". Making use of the fact that in an equitable game each player's mathematical expectation is zero, find the probability that A will win B's b dollars before he loses his a dollars.
A: no idea with this one thought about the binomial formula but realized that would not work.(Answer is supposed to be a/(a+b)
(Solved)Q 4.83: The amount of time(in minutes) that an executive of a certain firm talks on the telephone is a random variable having the probability density:
{ x/4 for 0<x<=2
f(x)={ 4/(x^3) for x>2{ 0 elsewhere
with reference to part (b) of Exercise 4.59, find the expected length of one of these telephone conversations that has lasted for 1 minute.A: The formula from the question is( E[u(x)|a<x<=b]= (aIntb(u(x)f(x)dx)) / (aIntb( f(x)dx))
I tried E[x|x>=1]= (1Int2(x*(x/4)dx)) / (1Int2((x/4)dx)) + (2Int(infinity)(x*(4/x^3)dx)) / (2Int(infinity)((4/x^3)dx))
=(14/6)/(9/6)+4=5.55555 minutes
but the back of the books says the answer is 2.95 mins so i don't know where i went wrong.
Essentially should be (j1+j2)/(j3+j4) not j1/j3+j2/j4
Any help would be appreciated.
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