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Step-by-Step Solution
Step 1: Understand the Given Conditions
We are provided with a set of 10 observations, denoted by $x_i$ for $i = 1, 2, \ldots, 10$, and the following information:
$\sum\limits_{i=1}^{10} \bigl(x_i - 5\bigr) = 10$
$\sum\limits_{i=1}^{10} \bigl(x_i - 5\bigr)^2 = 40$
We need to find the mean ($\mu$) and the variance ($\lambda$) of the shifted observations $x_1 - 3, x_2 - 3, \ldots, x_{10} - 3$.
Step 2: Simplify the First Summation
From
$$
\sum\limits_{i=1}^{10} \left(x_i - 5\right) = 10,
$$
we can rewrite it as
$$
\sum\limits_{i=1}^{10} x_i - \sum\limits_{i=1}^{10} 5 = 10.
$$
Since $\sum\limits_{i=1}^{10} 5 = 5 \times 10 = 50$, this gives:
$$
\sum\limits_{i=1}^{10} x_i - 50 = 10
\quad\Longrightarrow\quad
\sum\limits_{i=1}^{10} x_i = 60.
$$
Step 3: Simplify the Second Summation
From
$$
\sum\limits_{i=1}^{10} \bigl(x_i - 5\bigr)^2 = 40,
$$
expand each square:
$$
\bigl(x_i - 5\bigr)^2 = x_i^2 - 10x_i + 25.
$$
So,
$$
\sum\limits_{i=1}^{10} \bigl(x_i^2 - 10x_i + 25\bigr) = 40,
$$
which can be separated as
$$
\sum\limits_{i=1}^{10} x_i^2 - 10 \sum\limits_{i=1}^{10} x_i + 25 \times 10 = 40.
$$
We already know $\sum\limits_{i=1}^{10} x_i = 60$, so substitute that in:
$$
\sum\limits_{i=1}^{10} x_i^2 - 10 \times 60 + 250 = 40,
$$
thus
$$
\sum\limits_{i=1}^{10} x_i^2 = 40 + 600 - 250 = 390.
$$
Step 4: Compute the Mean (μ) of the Shifted Observations
The observations we are considering are $x_i - 3$ for $i = 1, 2, \ldots, 10$. Hence their mean is:
$$
\mu = \frac{(x_1 - 3) + (x_2 - 3) + \dots + (x_{10} - 3)}{10}.
$$
This simplifies to
$$
\mu = \frac{\sum\limits_{i=1}^{10} x_i - 3 \times 10}{10}
= \frac{60 - 30}{10}
= 3.
$$
Step 5: Compute the Variance (λ) of the Shifted Observations
The variance of $x_i - 3$ is given by
$$
\lambda = \frac{\sum\limits_{i=1}^{10} (x_i - 3)^2}{10} - \mu^2.
$$
First, expand $(x_i - 3)^2 = x_i^2 - 6x_i + 9$, so
$$
\sum\limits_{i=1}^{10} (x_i - 3)^2 = \sum\limits_{i=1}^{10} x_i^2
\;-\; 6\sum\limits_{i=1}^{10} x_i
\;+\; 9 \times 10.
$$
Substitute the known sums:
$$
\sum\limits_{i=1}^{10} x_i^2 = 390,
\quad
\sum\limits_{i=1}^{10} x_i = 60.
$$
So:
$$
\sum\limits_{i=1}^{10} (x_i - 3)^2 = 390 - 6 \times 60 + 9 \times 10
= 390 - 360 + 90
= 120.
$$
Therefore,
$$
\frac{\sum\limits_{i=1}^{10} (x_i - 3)^2}{10}
= \frac{120}{10}
= 12.
$$
Finally, subtract $\mu^2 = 3^2 = 9$ to get the variance:
$$
\lambda = 12 - 9 = 3.
$$
Step 6: Conclude the Ordered Pair (μ, λ)
From the above steps, we found:
The mean of the shifted observations is $\mu = 3$.
The variance of the shifted observations is $\lambda = 3$.
Hence, the ordered pair $(\mu, \lambda)$ is $(3, 3).$
