(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(4,31).
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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=31.
Step 2: Calculate P(r=1) for Poisson distribution
For a Poisson distribution, the probability mass function is given by P(r)=re−mmr!, 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)=1e−1(1)1!=e−1.
Step 3: Calculate mean for Binomial distribution
For a Binomial distribution B(n, p), the mean (μ) is given by the formula μ=np. Substituting the given values n=4 and p=31, we calculate the mean as 4×31=34.
Step 4: Calculate standard deviation for Binomial distribution
For a Binomial distribution, the variance (σ2) is npq, where q=1−p. The standard deviation (σ) is the square root of the variance. First, we find q=1−31=32. Then, we calculate the standard deviation as 4×31×32=98=322.