For the simple linear regression model Yi = β0 + β1 Xi +, ∼...

For the simple linear regression model $Y_i = \beta_0 + \beta_1 X_i + \epsilon_i$, $\epsilon_i \sim N(0, \sigma^2)$, for $i = 1, \dots, n$, show that $\text{Cov}(Y_i, \hat{Y}_i | X_i = x_i) = \sigma^2 \left(\frac{1}{n} + \frac{(x_i - \bar{x})^2}{\sum_{i=1}^n (x_i - \bar{x})^2}\right)$, where $\hat{Y}_i$ are the fitted values from the model. Clearly justify your steps. Hint: $\hat{\beta}_1 = \sum_{i=1}^n k_i Y_i$ where $k_i = \frac{x_i - \bar{x}}{\sum_{i=1}^n (x_i - \bar{x})^2}$.

Transcript

00:01 Hello everyone, in this question we have given one independent variable that is x and one dependent variable that is y.

00:08 Now we have given a set of pairs of values that is xy ,yi.

00:14 Now we need to calculate the best fit for the points we have given.

00:17 Now the set of values given for x is 3 ,9 ,11 ,5 ,2 and the set of points for the y is we have given 1 ,8 ,11 ,4 ,3.

00:38 Now we need to find two things.

00:40 First is we need to evaluate regression coefficients.

00:51 And second thing we need to find what is the value of the variable of y when x is equal to 7.

01:01 Ok, so first we need to know the formula for best fit line for linear regression is y is equal to a plus b x where a is equal to summation of y summation of x square minus summation of x summation of y whole divided by n times of summation x square minus summation of x whole of square and the value of b is n times of summation of x into y minus summation of x summation of y whole divided by n times of summation of x square minus summation of x of whole square and basically n is the number of data points.

02:08 Ok, now we have x that is 3 ,9 ,11 ,5 ,2 right and y as that is 1 ,8 ,11 ,4 ,3.

02:28 So if we calculate x of square we get 9, 81, 121, then 25 and then 4.

02:38 And if we calculate x into y we get 3, 72, 121, 20 and 6.

02:49 Now if we calculate summation of x that is we get this is equal to 30 when we add this column and then when we calculate summation of y then we get 27.

03:04 Similarly when we calculate summation of x square so we get 240 and similarly when we calculate summation of x y we get 222.

03:18 Ok now we can put all the values in the formula of a and b so we get a is equal to 27 times 240 minus 30 times of 222 whole divided by 5 times of 240 minus 30 of square right...

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