(9) Let X be a random variable that represents the number of tails before observing the 13-th head in repeated tosses of a coin. Assume the probability of observing a head on a single toss is p. Show that the density of X is (k + 13 – 1 P(X = k) = k = 0, 1, 2, … . 13 –1 )p"(1 - py*, The following are some proposed proofs. (a) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 heads and k tails, and the (k + 13)-th toss yielded a head. This happens with probability (k + 12) p12 (1 – p)* p. 12 P(X = k) = This simplifies out to the given statement. (b) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 tails and k heads, and the (k + 13)-th toss yielded a head. This happens with probability :+ 12) P(X = k) = - 12 This simplifies out to the given statement. (c) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 heads and k tails, and the (k + 13)-th toss yielded a head. This happens with probability (k+ 12) 12 )P*(1 – p)*p. P(X = k) = This simplifies out to the given statement. (d) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 tails and k heads, and the (k + 13)-th toss yielded a head. This happens with probability (k + 12) P(X = k) = p* (1 – 12 р. This simplifies out to the given statement. (e) None of the above The correct answer is (a) (b) (c) (d) (e) N/A (Select One)
(9) Let X be a random variable that represents the number of tails before observing the 13-th head in repeated tosses of a coin. Assume the probability of observing a head on a single toss is p. Show that the density of X is (k + 13 – 1 P(X = k) = k = 0, 1, 2, … . 13 –1 )p"(1 - py*, The following are some proposed proofs. (a) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 heads and k tails, and the (k + 13)-th toss yielded a head. This happens with probability (k + 12) p12 (1 – p)* p. 12 P(X = k) = This simplifies out to the given statement. (b) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 tails and k heads, and the (k + 13)-th toss yielded a head. This happens with probability :+ 12) P(X = k) = - 12 This simplifies out to the given statement. (c) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 heads and k tails, and the (k + 13)-th toss yielded a head. This happens with probability (k+ 12) 12 )P*(1 – p)*p. P(X = k) = This simplifies out to the given statement. (d) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 tails and k heads, and the (k + 13)-th toss yielded a head. This happens with probability (k + 12) P(X = k) = p* (1 – 12 р. This simplifies out to the given statement. (e) None of the above The correct answer is (a) (b) (c) (d) (e) N/A (Select One)
A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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Transcribed Image Text:(9) Let X be a random variable that represents the number of tails before observing the 13-th head in repeated tosses of a coin. Assume the probability of observing a head on
a single toss is p. Show that the density of X is
k + 13 – 1
P(X = k) =
1 – p)*,
k = 0, 1, 2, ....
13 –
- 1
The following are some proposed proofs.
(a) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 heads and k tails, and the (k + 13)-th
toss yielded a head. This happens with probability
k + 12`
P(X = k) = (* 2 )p"²(1 - p)* p-
This simplifies out to the given statement.
(b) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 tails and k heads, and the (k + 13)-th
toss yielded a head. This happens with probability
k + 12
P(X = k) =
12
This simplifies out to the given statement.
(c) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 heads and k tails, and the (k + 13)-th
toss yielded a head. This happens with probability
(*12 a -p*p.
+ 12)
p*(1 – p)'
P(X = k) =
This simplifies out to the given statement.
(d) Let k be a non-negative integer. The event {X = k} occurs only when over the earlier k + 12 tosses have observed exactly 12 tails and k heads, and the (k + 13)-th
toss yielded a head. This happens with probability
P(X = k) = (* 12 )*c1 – p" p.
+ 12`
\p*(1
This simplifies out to the given statement.
(e) None of the above
The correct answer is
(a)
(b)
(c)
(d)
(e)
N/A
(Select One)
Expert Solution
Step 1
Let X be a random variable such that
X: Number of tails before obtaining the 13th head in repeated tosses of a coin.
The probability of observing a head on a single toss is p and let k be a non-negative integer.
The event {X=k} occurs only when over the earlier k+12 tosses have exactly 12 heads and k tails, and the (k+13)th toss yielded a head. This happens with probability
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