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
- A biased six-faced die is tossed once. Suppose that the probability of any prime number showing up is twice that of any non-prime number showing up. Then, the probability that an odd number will show up is
- Let $X$ and $Y$ be independent Poisson random variables with means $4$ and $2$, respectively. Then, which of the following statements are true?
- Consider the M/M/1 queue in which customers arrive according to a Poisson process with rate $3$ and successive service times are independent exponential random variables having mean $\frac{1}{9}$. Let $P_n$ be the long run probability that there are exactly $n$ customers in the system. Then, which of the following statements are true?
- 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?
