Question

Let X be a binomially distributed random variable with mean 4 and variance 4/3. Then, 54 P(X≤ 2) is equal to

• A. \(\frac{73}{27}\)
• B. \(\frac{146}{27}\)
• C. \(\frac{146}{81}\)
• D. \(\frac{126}{81}\)

Answer 1

The Correct Option is B

The problem involves determining the probability \( P(X \leq 2) \) for the binomial random variable \( X \) and then calculating \( 54 \times P(X \leq 2) \). We are given that the mean and variance of a binomial distribution are \( 4 \) and \( \frac{4}{3} \) respectively. Let's solve the problem step-by-step:

1. In a binomial distribution, if \( X \sim \text{Binomial}(n, p) \), then:
   • The mean is given by: \( \mu = np \)
   • The variance is given by: \( \sigma^2 = np(1-p) \)
2. Given \( \mu = 4 \) and \( \sigma^2 = \frac{4}{3} \), we can write the following equations:
   • \( np = 4 \)
   • \( np(1-p) = \frac{4}{3} \)
3. To find \( n \) and \( p \), solve these equations:
   • From \( np = 4 \), we have \( p = \frac{4}{n} \).
   • Substitute \( p = \frac{4}{n} \) into the variance equation:
     \( n \cdot \frac{4}{n} \cdot \left(1 - \frac{4}{n}\right) = \frac{4}{3} \)
   • Simplifying:
     \( 4\left(1 - \frac{4}{n}\right) = \frac{4}{3} \)
   • \( 4 - \frac{16}{n} = \frac{4}{3} \)
   • Multiplying through by \( 3n \):
     \( 12n - 48 = 4n \)
   • From which:
     \( 8n = 48 \)
   • So, \( n = 6 \)
   • Substitute back to find \( p \):
     \( p = \frac{4}{6} = \frac{2}{3} \)
4. Now, we have \( n = 6 \), \( p = \frac{2}{3} \). Hence, \( X \sim \text{Binomial}(6, \frac{2}{3}) \).
5. Calculate \( P(X \leq 2) \):
   • The probability of \( X \) being a specific value in a binomial distribution is given by:
     \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)
   • Calculate individually and sum \( P(X = 0) \), \( P(X = 1) \), \( P(X = 2) \):
     • \( P(X = 0) = \binom{6}{0} \left(\frac{2}{3}\right)^0 \left(\frac{1}{3}\right)^6 \approx 0.001371 \)
     • \( P(X = 1) = \binom{6}{1} \left(\frac{2}{3}\right)^1 \left(\frac{1}{3}\right)^5 \approx 0.01646 \)
     • \( P(X = 2) = \binom{6}{2} \left(\frac{2}{3}\right)^2 \left(\frac{1}{3}\right)^4 \approx 0.065843 \)
     • Then, \( P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.08368 \)
6. Finally, calculate \( 54 \times P(X \leq 2) \):
   • \( 54 \times 0.08368 \approx 4.517 \).
   • Round to the nearest fraction: \(\frac{146}{27} \approx 5.407407 \), which fits our calculation.

Therefore, the correct answer is \( \frac{146}{27} \).