Question

chegg jon flips a fair coin until a head is witnessed, and independently anna flips a fair coin (a different coin than jon’s) until a head is witnessed. let w be the minimum of the number of flips between john and anna. (a) derive the pmf for w. (b) compute e(w). (c) derive an expression for p(w ≥ k) for k

Answer 1

The probability mass function of for W is given by $P(W=K)=\frac{3}{2^{2K}}$, the expected value(e(w)) is  $\frac{4}{3}$ and the expressions for $P(W\geq K)$ is given by $\frac{1}{2}^{2k-2},K=1,2,3...$

To solve this problem we have to understand pmf or probability mass function and e(W) or expected value.

Probability mass function is defined as the function which given the probability of a discreet random variable.

The weighted average of the possible values of a random variable with weights given by their respective probabilities is known as expected value.

Jon flips a fair coin until a head is witnessed, and independently Anna Flips a fair coin(a different coin than Jon) until a head is witnessed. Let W be the number of flips between Jon and Anna.

Let us now define the random variable

X: Number of flips done by Jon

Y: Number of flips done by Mocus

So, PMF of X

$P(X=x)=\frac{1}{2}\times(1-\frac{1}{2})^{(x-1)},x=1,2,...\\P(X=x)=(\frac{1}{2})^x$

$P(X\geq x)=\sum_{i=x}^{\infty}(\frac{1}{2})^i\\P(X\geq x)=(\frac{1}{2})^x+(\frac{1}{2})^{x+1}+.....\\P(X\geq x)=\frac{(\frac{1}{2})^x}{1-\frac{1}{2}}\\P(X\geq x)=(\frac{1}{2})^{x-1}$

Simmilarly

$P(Y\geq y)=(\frac{1}{2})^{(y-1)},y=1,2,3....$

Now W=min(X,Y)

So,

$P(w\geq K)\\=P(min(X,Y)\geq K)\\=P(X\geq K,Y\geq K)\\=(\frac{1}{2})^{K-1}(\frac{1}{2})^{K-1}\\=(\frac{1}{2})^{2K-2}$

a) Now let us derive the probability mass function(pmf) for W:

$P(W=K)=P(W\geq K)-P(W\geq K+1)\\P(W=K)=\frac{3}{2^{2K}}$

b) Let us calculate the expected value or e(W)

$E(W)=\sum^{\infty}_{i=1}ip(W=0)$

Solving we get

$=\sum^{\infty}_{I=1}P(W\geq i)\\=\sum^{\infty}_{I=1}\frac{1}{2^{2i-2}}\\=1+\frac{1}{4}+\frac{1}{16}+....\\=\frac{4}{3}$

c) The expression for $P(W\geq K)$ for K=1,2,3,4....

$P(W\geq K)\\=\frac{1}{2}^{2k-2},K=1,2,3...$

To learn more about probability mass function:

brainly.com/question/14263946

To learn more about expected value:

brainly.com/question/28197299

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