MATH1715 Tutee Revision Questions

Response: Doodle! Draw a picture illustrating some examples! For example, HT, HHT,
HHHT, HHHHT, TH, TTH, TTTH, and so on. It looks like a series of Bernoulli trials
which stop when something different (a success) occurs. This should lead you to consider a
geometric distribution, the distribution of the time to the first success. The main difference
between this and the usual geometric distribution is that a “success” can be either H or T
and is determined by the first toss. You need an initial toss to determine what the “success”
is going to be!
Let X denote the number of tosses required to see both sides once. We require E[X].
We toss the coin once and see side A say. Let A now denote a Failure, and the other side
B a Success. The probability of a Success is p = 0.5. Tosses are independent. A toss either
gives side A, a failure, or side B, a success. What is the distribution of the number of
(subsequent) tosses Y until the first Success? (Here Y takes values 1, 2, 3, ....) Clearly
X = Y + 1, so E[X] can be found.
An alternative approach is to determine the distribution of X directly and then calculate
E[X] = �
x P{X = x} .
Thus
P{X = 1} = 0
P{X = 2} = P{HT or TH} = 0.25 + 0.25 = 0.5
x
P{X = 3} = P{HHT or TTH} = 0.125 + 0.125 = 0.25
P{X = 4} = 0.125
and so on. Writing Y = X − 1, then Y has the usual geometric distribution
P{Y = 1} = P{X = 2} = 0.5
P{Y = 2} = P{X = 3} = 0.5 2
P{Y = 3} = P{X = 4} = 0.5 3
and so on. Since E[Y ] = 1/0.5 = 2, then E[Y ] = E[X − 1] = E[X] − 1 gives E[X] = 3.
A simple R program would let you check my answer.
x=numeric(100000) # Initialise the counts of x to zero.
for (i in 1:100000){ # Do 100000 simulations.
u=rbinom(1,1,0.5) # Generate a random Bin(n=1,p=0.5) value.
# u=0 gives a Tail and u=1 gives a Head.
# Now generate tosses 2,3,4,5,.... until we get something not equal to u.
x[i]=1 # For simulation i initialise the count of the number of tosses (1 so far!).
v=-1 # v denotes subsequent tosses, set to -1 to start with.
# This is just to give R a value v for the next line.
while (v != u){ # While v does not equal u keep on tossing the coin.
v=rbinom(1,1,0.5) # Generate subsequent tosses, v=0 is a T and v=1 is a H say.
x[i]=x[i]+1
}
} # End the 100000 simulations.
mean(x) # Mean of x.
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