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# Problem Set 14 Balls in Bins Estimation

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EECS 126: Probability and Random Processes
Problem Set 14 (Optional)

1. Balls in Bins Estimation
We throw n ≥ 1 balls into m ≥ 2 bins. Let X and Y represent the number of balls that land
in bin 1 and 2 respectively.
(a) Calculate E[Y | X].
(b) What are L[Y | X] and Q[Y | X] (where Q[Y | X] is the best quadratic estimator of Y
given X)?
Hint: Your justification should be no more than two or three sentences, no calculations
necessary! Think carefully about the meaning of the MMSE.
E[X] and E[Y ].
(d) Compute var(X).
(e) Compute cov(X, Y ).
(f) Compute L[Y | X] using the formula. Ensure that your answer is the same as your
2. MMSE and Conditional Expectation
Let X, Y1, . . . , Yn be square integrable random variables. The MMSE of X given (Y1, . . . , Yn)
is defined as the function φ(Y1, . . . , Yn) which minimizes the mean square error
E[(X − φ(Y1, . . . , Yn))2
].
(a) For this part, assume n = 1. Show that the MMSE is precisely the conditional expectation
E[X|Y ]. Hint: expand the difference as (X − E[X|Y ] + E[X|Y ] − φ(Y )).
(b) Argue that
E

(X − E[X | Y1, . . . , Yn])2

≤ E
X −
1
n
Xn
i=1
E[X | Yi
]
2
.
That is, the MMSE does better than the average of the individual estimates given each
Yi
.
3. Geometric MMSE
Let N be a geometric random variable with parameter 1 − p, and (Xi)i∈N be i.i.d. exponential
random variables with parameter λ. Let T = X1 + · · · + XN . Compute the LLSE and MMSE
of N given T.
Hint: Compute the MMSE first.
1

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