Suppose that Y1 and Y2 are binomial random variables with

QUESTION:

 Suppose that \(Y_{1}\) and \(Y_{2}\) are binomial random variables with parameters \(\left(n, p_{1}\right)\) and \(\left(n, p_{2}\right)\), respectively, where \(p_{1}<p_{2}\). (Note that the parameter  is the same for the two variables.)
a Use the binomial formula to deduce that \(P\left(Y_{1}=0\right)>P\left(Y_{2}=0\right)\).
b Use the relationship between the beta distribution function and sums of binomial probabilities given in Exercise 4.134 to deduce that, if  is an integer between 1 and \(n-1\),

                    \(P\left(Y_{1} \leq k\right)=\sum_{i=0}^{k}\left(\begin{array}{l}

n \\

i

\end{array}\right)\left(p_{1}\right)^{i}\left(1-p_{1}\right)^{n-i}=\int_{p_{1}}^{1} \frac{t^{k}(1-t)^{n-k-1}}{B(k+1, n-k)} d t

\).

c If  is an integer between 1 and \(n-1\), the same argument used in part (b) yields that

                  \(P\left(Y_2\le k\right)=\sum_{i=0}^k\left(\begin{array}{l}n\\

i\end{array}\right)\left(p_2\right)^i\left(1-p_2\right)^{n-i}=\int_{p_2}^1\frac{t^k(1-t)^{n-k-1}}{B(k+1,\ n-k)}dt.

\)

Show that, if  is any integer between 1 and \(n-1\), \(P\left(Y_{1} \leq k\right)>P\left(Y_{2} \leq k\right)\). Interpret this result.

Equation Transcription:

Text Transcription:

Y_1

Y_2

(n,p_1)

(n,p_2)

p_1<p_2

P(Y_1=0)>P(Y_2=0)

n-1

P(Y_1</=k)=sum i=0 k(_i^n)(p_1)^i(1-p_1)^n-i=integral p1 to 1 t^k(1-t)^n-k-1 over B(k+1, n-k)dt

n-1

P(Y_</=2k)=sum i=0 k(_i^n)(p_2)^i(1-p_2)^n-i=integral p2 to 1 t^k(1-t)^n-k-1 over B(k+1, n-k)dt

n-1

P(Y_1</=k)>P(Y_2</=k)