For a Poisson distribution, the probability mass function is given by P(r) = (e^(-m) m^r)/(r!), where m is the mean and r is the number of occurrences. Substituting m=1 and r=1 into the formula, we get P(r=1) = (e^(-1) (1)^1)/(1!) = e^(-1).

For a Poisson distribution, the probability mass function is given by P(r) = (e^(-m) m^r)/(r!), where m is the mean and r is the number of occurrences. Substituting m=1 and r=1 into the formula, we get P(r=1) = (e^(-1) (1)^1)/(1!) = e^(-1).

For a Binomial distribution B(n, p), the mean () is given by the formula = np. Substituting the given values n=4 and p=(1)/(3), we calculate the mean as 4 × (1)/(3) = (4)/(3).

For a Binomial distribution, the variance ( ²) is npq, where q = 1-p. The standard deviation () is the square root of the variance. First, we find q = 1 - (1)/(3) = (2)/(3). Then, we calculate the standard deviation as √(4 × (1)/(3) × (2)/(3)) = √((8)/(9)) = (2√2)/(3).

For a Binomial distribution, the variance ( ²) is npq, where q = 1-p. The standard deviation () is the square root of the variance. First, we find q = 1 - (1)/(3) = (2)/(3). Then, we calculate the standard deviation as √(4 × (1)/(3) × (2)/(3)) = √((8)/(9)) = (2√2)/(3).

(a) For a Poisson distribution, if mean (m) = 1, then find P( $r = 1$ ). OR (b) Find the mean and standard deviation of the Binomial distribution $B(4, \frac{1}{3})$ .

Answer: (a) $P(r=1) = e^{-1}$ (approximately 0.3679) \\ (b) Mean = $\frac{4}{3}$ , Standard Deviation = $\frac{2\sqrt{2}}{3}$

Step-by-step solution

Step 1: Identify the problem type and given values

The problem presents two independent parts. We will solve both. Part (a) involves a Poisson distribution with a given mean $m=1$ . Part (b) involves a Binomial distribution with parameters $n=4$ and $p=\frac{1}{3}$ .

Step 2: Calculate P( $r=1$ ) for Poisson distribution

For a Poisson distribution, the probability mass function is given by $P(r) = \frac{e^{-m} m^r}{r!}$ , where $m$ is the mean and $r$ is the number of occurrences. Substituting $m=1$ and $r=1$ into the formula, we get $P(r=1) = \frac{e^{-1} (1)^1}{1!} = e^{-1}$ .

Step 3: Calculate mean for Binomial distribution

For a Binomial distribution $B(n, p)$ , the mean ( $\mu$ ) is given by the formula $\mu = np$ . Substituting the given values $n=4$ and $p=\frac{1}{3}$ , we calculate the mean as $4 \times \frac{1}{3} = \frac{4}{3}$ .

Step 4: Calculate standard deviation for Binomial distribution

For a Binomial distribution, the variance ( $\sigma^2$ ) is $npq$ , where $q = 1-p$ . The standard deviation ( $\sigma$ ) is the square root of the variance. First, we find $q = 1 - \frac{1}{3} = \frac{2}{3}$ . Then, we calculate the standard deviation as $\sqrt{4 \times \frac{1}{3} \times \frac{2}{3}} = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}$ .

Solve your own maths question free →

(a) For a Poisson distribution, if mean (m) = 1, then find P(r=1r = 1r=1). OR (b) Find the mean and standard deviation of the Binomial distribution B(4,13)B(4, \frac{1}{3})B(4,31​).