Consider a deck consisting of seven cards, marked 1, 2, 7. three of these cards are selected at random. define an rv w by w ¼ the sum of the resulting numbers, and compute the pmf of w. then compute e(w) and var(w). [hint: consider outcomes as unordered, so that (1, 3, 7) and (3, 1, 7) are not different outcomes. then there are 35 outcomes, and they can be listed.] (this type of rv actually arises in connection with wilcoxon’s rank-sum test, in which there is an x sample and a y sample and w is the sum of the ranks of the x’s in the combined sample.)

There are possible outcomes. At the least, you can draw (1, 2, 3) to get a sum of 6, and at most (5, 6, 7) for a sum of 18. Then the PMF is

That is, consider all the possible sums and their integer partitions. For example,

6 = 1 + 2 + 3

7 = 1 + 2 + 4

8 = 1 + 2 + 5 = 1 + 3 + 4

9 = 1 + 2 + 6 = 1 + 3 + 5 = 2 + 3 + 4

and so on.

Then the expected value is

We can find the variance via

Good morning ☕️

Step-by-step explanation:

NOTE: The idea here is this:

If two pairs of corresponding angles in ΔABC and  ΔGEC are congruent, then

ΔABC ∼ ΔGEC.

since ΔABC ≅ ΔDEF then ∠ABC and ∠DEF are congruent (1)

and since the pair of angles ∠DEF and ∠CEG is vertical then ∠DEF and

∠CEG are congruent (2)

from the relations (1) and (2) we conclude that ∠ABC and ∠CEG are congruent  (3)

now, the pair of angles ∠ACB and ∠ECG is vertical then ∠ACB and

∠ECG are congruent (4)

NOTE + (3) + (4) prove that ΔABC is similar to  ΔGEC

:)