Discussion
If the probability generating function of a random variable X is given by :
\[ P_{X}(t) = (0.4 + 0.6t)^{10}, \]
then the distribution of X is :
Answer: B
X is binomial with mean 6.
The probability generating function of a binomial \((n, p)\) variable is \((q + pt)^{n}\) with \(q = 1 - p\). Matching \((0.4 + 0.6t)^{10}\) gives \(n = 10\), \(p = 0.6\), \(q = 0.4\), so
\[ \begin{aligned} E(X) &= np = 10 \times 0.6 = 6 \\ P_{X}'(1) &= 10 \times 0.6 \times 1^{9} = 6 \end{aligned} \]
Mean 4 is \(nq\), the expected number of failures, which comes from reading 0.4 as the success probability. A negative binomial pgf has the form \(p^{r}(1 - qt)^{-r}\), with a negative exponent, so neither negative binomial option fits a polynomial pgf.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development