4. Suppose a statistician is bored one afternoon in his office, and decides to t

Question

4. Suppose a statistician is bored one afternoon in his office, and decides to toss a coin until he tosses heads. Before he starts, he uses his knowledge of probability theory to deduce the following probability model: Tosses Probability .5 .25 .125 0625 0313 .0156 0078 0078 There is only a limited amount of time before class, so if the statistician has not tossed heads after the first seven trials, he puts the coin back in his pocket and just uses the "gambler's fallacy" to stipulate that the eighth toss would have been heads. a) Calculate the mean number of tosses it should take to get his first head. b) Calculate the variance of this random variable. c) Calculate the standard deviation of this random variable 5. Suppose a statistician is bored one afternoon in her office, and decides to roll a tetrahedron until she rolls an A (assume the sides are lettered A, B, C, and D). Before she starts, she uses her knowledge of probability theory to deduce the following probability model Rolls 6 Probability 25 1875 1406 1055 .0791 0593 0445 1335 There is only a limited amount of time before class, so if the statistician has not rolled an A after the first seven trials, she puts the tetrahedron back in her desk and just relies on the "gambler's fallacy" to stipulate that the eighth roll would have been an A. a) b) c) Calculate the mean number of rolls it should take to get her first A. Calculate the variance of this random variable. Calculate the standard deviation of this random variable.

Explanation / Answer

SolutionA

mean=totalxP(X=x)

=1*0.5+2*0.25+3*.125+4*0.0625+5*0.0313+6*0.0156+7*0.0078+8*0.0078

=1.9921

mean=E(x)=1.9921

variance=tota x^2*P(X=x)-E(X)^2

=1^2*0.5+2^2*0.25+3^2*.125+4^2*0.0625+5^2*0.0313+6^2*0.0156+7^2*0.0078+8^2*0.0078-(1.9921)^2

=5.8505-3.968462

=1.882038

variance=1.882038

standard deviation=sqrt(variance)

=sqrt(1.882038)

=1.371874

std dev=1.371874

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