Investment Strategy for Maximum Expected Dividend
To maximize the expected dividend from an investment of Rs. 1000, we compare the expected return for each proposed allocation between Company I and Company II, considering market conditions and probabilities.
Expected Dividend Calculation per Option
Option 1: Rs. 1000 in Company I
Investment: Rs. 1000 in Company I.
- Dividend in good market (Prob 0.4): $1000 \times 50\% = 1000 \times 0.50 = 500$.
- Dividend in bad market (Prob 0.6): $1000 \times 10\% = 1000 \times 0.10 = 100$.
- Expected Dividend: $(0.4 \times 500) + (0.6 \times 100) = 200 + 60 = 260$.
Option 2: Rs. 1000 in Company II
Investment: Rs. 1000 in Company II.
- Dividend in good market (Prob 0.4): $1000 \times 30\% = 1000 \times 0.30 = 300$.
- Dividend in bad market (Prob 0.6): $1000 \times 20\% = 1000 \times 0.20 = 200$.
- Expected Dividend: $(0.4 \times 300) + (0.6 \times 200) = 120 + 120 = 240$.
Option 3: Rs. 500 in Company I and Rs. 500 in Company II
Investment: Rs. 500 in Company I and Rs. 500 in Company II.
- Dividend in good market (Prob 0.4): $(500 \times 50\%) + (500 \times 30\%) = (500 \times 0.50) + (500 \times 0.30) = 250 + 150 = 400$.
- Dividend in bad market (Prob 0.6): $(500 \times 10\%) + (500 \times 20\%) = (500 \times 0.10) + (500 \times 0.20) = 50 + 100 = 150$.
- Expected Dividend: $(0.4 \times 400) + (0.6 \times 150) = 160 + 90 = 250$.
Option 4: Rs. 600 in Company I and Rs. 400 in Company II
Investment: Rs. 600 in Company I and Rs. 400 in Company II.
- Dividend in good market (Prob 0.4): $(600 \times 50\%) + (400 \times 30\%) = (600 \times 0.50) + (400 \times 0.30) = 300 + 120 = 420$.
- Dividend in bad market (Prob 0.6): $(600 \times 10\%) + (400 \times 20\%) = (600 \times 0.10) + (400 \times 0.20) = 60 + 80 = 140$.
- Expected Dividend: $(0.4 \times 420) + (0.6 \times 140) = 168 + 84 = 252$.
Investment Choice for Maximum Expected Dividend
Comparing the expected dividends:
- Option 1: Rs. 260
- Option 2: Rs. 240
- Option 3: Rs. 250
- Option 4: Rs. 252
The highest expected dividend is Rs. 260, achieved by investing Rs. 1000 in Company I.
Important Questions from Discrete Probability
- Let $X$ be a Binomial$(n, p)$ random variable, where $n \in \{5,6\}$ and $p\in \{\frac{1}{4}, \frac{3}{4}\}$. If $X = 3$ is observed, then the maximum likelihood estimate of $(n, p)$ is
- Suppose two fair dice are thrown independently at random. Let $X$ and $Y$ be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?
- A box contains 40 numbered red balls and 60 numbered black balls. From the box, balls are drawn one by one at random without replacement till all the balls are drawn. The probability that the last ball drawn is black equals
Consider the problem of testing $H_0 : \theta = 1$ vs $H_1 : \theta = \frac{1}{2}$ where $\theta$ is the mean of a Poisson random variable. Let $X$ and $Y$ be a random sample from Poisson ($\theta$) distribution. Consider the following test procedure:
Reject $H_0$ if either $X = 0$ or $(X = 1 \text{ and } X + Y \leq 2)$; otherwise accept $H_0$.
Which of the following are true?
In a football league, the goals scored by home teams over 380 matches have the following frequency distribution.
Number of goals 0 1 2 3 4 5 Frequency 92 121 91 50 19 7 The average goals scored by home teams is 1.49. We want to test $H_0$: Goal distribution is Poisson. Based on observations the value of the $\chi^2$-statistic for goodness of fit is 1.27. Given $\chi^2_{0.05, 6} = 1.64, \chi^2_{0.05, 5} = 1.15, \chi^2_{0.95, 6} = 12.59$ and $\chi^2_{0.95, 5} = 11.07$, which of the following are true?
