Question

Let \(X\) be a discrete random variable. If \[ P(X=r)=\frac{3}{7}a^r,\qquad r=0,1,2,\ldots,\infty,\quad 0<a<1, \] then \(aP(X=2)=\)

• A. \(P(X=1)\)
• B. \(\dfrac37P(X=2)\)
• C. \(P(X=3)\)
• D. \(\dfrac47P(X=1)\)

Hint:

When a probability mass function is of the form \[ \boxed{P(X=r)=ka^r,} \] multiplying \(P(X=r)\) by \(a\) gives \[ \boxed{aP(X=r)=P(X=r+1).} \]

Answer 1

The Correct Option is C

Step 1: Find \(P(X=2)\). Given, \[ P(X=r)=\frac37a^r. \] Hence, \[ P(X=2)=\frac37a^2. \]

Step 2: Multiply by \(a\). Therefore, \[ aP(X=2) = a\left(\frac37a^2\right) = \frac37a^3. \]

Step 3: Compare with the given probability mass function. Since \[ P(X=3)=\frac37a^3, \] we get \[ aP(X=2)=P(X=3). \] Hence, \[ \boxed{P(X=3)}. \] Thus, \[ \boxed{(C)} \] is the correct answer.