KCET · Maths · Probability
A random variable \(X\) has the following probability distribution:
If the mean of the random variable \(X\) is \(1 / 3\), then the variance is
- A \(\frac{1}{18}\)
- B \(\frac{5}{18}\)
- C \(\frac{7}{18}\)
- D \(\frac{11}{18}\)
Answer & Solution
Correct Answer
(B) \(\frac{5}{18}\)
Step-by-step Solution
Detailed explanation
\(\because\) Mean \(=\sum_{i=1}^n x_i P\left(x_i\right)\)
\(\Rightarrow \quad \frac{1}{3}=0 \times \frac{25}{36}+1 \times k+2 \times \frac{1}{36}\)
\(\Rightarrow \quad \frac{1}{3}=k+\frac{1}{18}\)
\(\therefore \quad k=\frac{1}{3}-\frac{1}{18}=\frac{6-1}{18}=\frac{5}{18}\)
Now, variance \(=\sum_{i=0}^n x_i^2 P_i(x)-(\text { mean })^2\)
\(=(0)^2 \times \frac{25}{36}+(1)^2 \times k+(2)^2 \times \frac{1}{36}-\frac{1}{9}\)
\(=0+k+\frac{1}{9}-\frac{1}{9}=k=\frac{5}{18}\)
Hence, variance \(=\frac{5}{18}\)
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