Let X1,X2,X3 denote a random sample of size n = 3 from a

QUESTION:

Let \(X_{1}, X_{2}, X_{3}\) denote a random sample of size \(n=3\) from a distribution with the geometric pmf

\(f(x)=\left(\frac{3}{4}\right)\left(\frac{1}{4}\right)^{x-1}, \quad x=1,2,3, \ldots\).

(a) Compute \(P\left(X_{1}=1, X_{2}=3, X_{3}=1\right)\).

(b) Determine \(P\left(X_{1}+X_{2}+X_{3}=5\right)\).

(c) If \(Y\) equals the maximum of \(X_{1}, X_{2}, X_{3}\), find

\(P(Y \leq 2)=P\left(X_{1} \leq 2\right) P\left(X_{2} \leq 2\right) P\left(X_{3} \leq 2\right) \text {. }\)

Equation Transcription:

Text Transcription:

X_1,X_2,X_3  

n=3

f(x)=(¾)(¼)^x-1, x=1,2,3,….  

P(X_1=1,X_2=3, X_3=1)  

P(X_1+X_2+X_3=5)  

Y  

P(Y≤2)=P(X_1 < or = 2)P(X_2 < or = 2)P(X_3 < or =2)