a) The pmf for w for certain value n is P(W = n) = 3 / 2²ⁿ⁺², b) The expected value on w, E(w) = 4 / 3, c) The expression for P(w ≥ k) = , where k = 1, 2, 3...
Given: Jon flips a fair coin until a head is witnessed.
Anna flips a fair coin independently until a head is witnessed.
What is the probability density function?
The probability density function is a function that determines the probability for a discrete random variable X over a given sample space S for a certain value for X = x(some value).
It is also denoted as pmf and is written as P(X = x).
Let's solve the given question:
Let us assume the number of flips Jon did until he got a head to be X.
Also as Anna's flip independently so we need to consider a different random variable say Y.
So, the probability mass function for Jon (X) for a certain value of x is:
P(X = x) = , where x = 1, 2, 3, ...........
P(X = x) =
P(X = x) =
P(X = x) = , where x = 1, 2, 3, ........
Similarly, for Anna, we will have the same probability mass function but with a different random variable Y for a certain value of y
P(Y = y) = , where y = 1, 2, 3, ........
Now it is given that, w is the minimum number of flips between Jon and Anna.
w = minimum(P(X = x), P(Y = y))
Let us suppose the probability mass distribution over w is n for a certain value.
Then
P(w ≥ n) = P(minimum(P(X = x), P(Y = y)) ≥ n)
=
=
Therefore, the probability mass function for w is
P(W = n) = P(W ≥ n) - P(W ≥ n + 1)
= -
= -
= ( 1 -
)
= ( 1 - 1 / 4)
= (4 - 1) / 4
= 3 / 4 ()
= 3 / 2² ()
= 3 / 2²ⁿ⁺²
Now the expected value of w, E(w) is:
E(w) = ∑P(w ≥ i) where i = 0 to ∞
= ∑<em> </em>where i = 0 to ∞
= 1 + 1 / 4 + 1 / 16 + .......
This is infinite GP series. So the summation of infinite GP is
S = a / ( 1 - r )
where a is the first term, r is the power and s is the summation.
Here a = 1, r = 1 / 4
S = 1 / (1 - 1 / 4)
S = 1 / 3 / 4
S = 4 / 3
Therefore, E(w) = 4 / 3
The expression for P(w ≥ k) is:
P(w ≥ k) = , where k = 1, 2, 3...
Hence
a) The pmf for w for certain value n is P(W = n) = 3 / 2²ⁿ⁺²
b) The expected value on w, E(w) = 4 / 3
c) The expression for P(w ≥ k) = , where k = 1, 2, 3...
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