Respuesta :
Answer and explanation:
Given : A fair coin will be tossed 200,000 times. Let X denote the number of Tails.
To find :
(a) What is the expected value and the standard deviation of X?
Given that total number of tosses is n=200000
Probability of getting tail in single toss is p=0.5
Expected value is given by,
[tex]E(X)=n\times p[/tex]
[tex]E(X)=200000\times 0.5[/tex]
[tex]E(X)=100000[/tex]
Standard deviation is given by,
[tex]SD=\sqrt{n\times p\times q}[/tex]
[tex]SD=\sqrt{200000\times 0.5\times 0.5}[/tex]
[tex]SD=\sqrt{50000}[/tex]
[tex]SD=223.60[/tex]
(b) Consider a game in which you have to pay $5 in order to earn [tex]\$\log_{10}(X)[/tex] when X > 0. Is this a fair game? If not, your expected profit is positive or negative?
We have to pay $5 to get [tex]\$\log_{10}(X)[/tex]
Minimum number of tails required to get $5 is 100000 .
Since, we get X=100000 with Probability 0.5 and for winning we need more number of tosses.
Probability of losing is more than profit hence it's biased test .
As expected number of tails =100000
So profit is given by,
[tex]P=\$\log_{10}(100000)-5[/tex]
[tex]P=\$\log_{10}(10^5)-5[/tex]
[tex]P=5-5[/tex]
[tex]P=0[/tex]
Therefore, The profit is zero.
