JEE MAIN - Mathematics (2024 - 9th April Evening Shift - No. 13)
If the variance of the frequency distribution
| $$x$$ | $$c$$ | $$2c$$ | $$3c$$ | $$4c$$ | $$5c$$ | $$6c$$ |
|---|---|---|---|---|---|---|
| $$f$$ | 2 | 1 | 1 | 1 | 1 | 1 |
is 160, then the value of $$c\in N$$ is
5
8
6
7
Explanation
| $$x_i$$ | $$f(x_i)$$ | $$x(f(x)$$ | $$x^2f(x)$$ |
|---|---|---|---|
| C | 2 | 2C | 2C$$^2$$ |
| 2C | 1 | 2C | 4C$$^2$$ |
| 3C | 1 | 3C | 9C$$^2$$ |
| 4C | 1 | 4C | 16C$$^2$$ |
| 5C | 1 | 5C | 25C$$^2$$ |
| 6C | 1 | 6C | 36C$$^2$$ |
$$\begin{aligned} & \sigma^2=E\left(x^2\right)-[E(x)], \sum f\left(x_i\right)=7 \\ & E(x)=\sum x f(x)=22 C \\ & E\left(x^2\right)=\sum x^2 f(x)=92 C^2 \end{aligned}$$
$$\begin{aligned} & \sigma^2=160=\frac{92 C^2}{7}-\left(\frac{22 C}{7}\right)^2 \\ & \Rightarrow C= \pm 7 \text { but } C \in N \\ & \Rightarrow C=7 \end{aligned}$$
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