# Probability

Question:A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K=5. For K=1,2,3...K, let Xk be a continuous random variable that is uniform over the interval [0,5] . The Xk are independent of one another and of the coin flips. Let X = ∑ Xk (from k=1 to K). Find the mean and variance of X . You may use the fact that the mean and variance of a geometric random variable with parameter p are 1/p and (1-p)/p^2, respectively.

My attempt:
E[N] = 1/p = 2
E[Xk] = 1/2 *5 = 5/2
E[X] = E[N]*E[Xk]
=2 * 5/2 = 5
Var[N] = (1-p)/p^2 = 2
var[Xk] = 1/12 * 5^2 = 25/12
Var[X] = E[N]var[Xk] + (E[Xk]^2)(var[N])
=....

But both my E[X] and Var[X] gave wrong answers :(

1. 👍
2. 👎
3. 👁
1. Ok, I know my mistake, I didn't read the question carefully, but was on the right track

1. 👍
2. 👎
2. How did you take into account the "plus 1"?

1. 👍
2. 👎
3. Firstly, I assume N=K in your solutions. The expected value and variance of X can be found via Law of Iterated Expectation (LIE) and Law of Total Variance (LTV):

E[X]=E[E[X|K]], var(X)=E[var(X|K)]+var(E[X|K])

For the expectation, your approach is correct, but it can be found via LIE:
E[X|K]=KE[Xk]→E[KE[Xk]]=E[K]E[Xk]
You just need to correct your expectation for K: E[K]=1/p+1, since it is of the form 1+Y, where Y is a geometric RV with parameter p. Also, note that var(K)=var(1+Y)=var(Y)=(1−p)/p2 as yours.

For the variance, we need var(X|K)=var(∑Xk|K)=Kvar(Xk), and by LTV:
var(X)=E[Kvar(Xk)]+var(KE[Xk])=var(Xk)E[K]+E[Xk]2var(K)

Substituting:

E[X] = 15/2
Var[X] = 18.75

1. 👍
2. 👎

## Similar Questions

1. ### Statistics

You are shown a coin that its owner says is fair in the sense that it will produce the same number of heads and tails when flipped a very large number of times. Suppose you decide to flip a coin 100 times. a. What conclusion would

2. ### Probability

A fair coin is tossed repeatedly and independently. We want to determine the expected number of tosses until we first observe Tails immediately preceded by Heads. To do so, we define a Markov chain with four states, {S,H,T,HT},

3. ### Probability Theory

A fair coin is tossed three times and the events A, B, and C are defined as follows: A:{ At least one head is observed } B:{ At least two heads are observed } C:{ The number of heads observed is odd } Find the following

4. ### probability

You flip a fair coin (i.e., the probability of obtaining Heads is 1/2) three times. Assume that all sequences of coin flip results, of length 3, are equally likely. Determine the probability of each of the following events. {HHH}:

1. ### probability, mathematics, statistics

The probability of Heads of a coin is y , and this bias y is itself the realization of a random variable Y which is uniformly distributed on the interval [0,1] . To estimate the bias of this coin. We flip it 6 times, and define

2. ### Statistics

Suppose that X , Y , and Z are independent, with E[X]=E[Y]=E[Z]=2 , and E[X2]=E[Y2]=E[Z2]=5 . Find cov(XY,XZ) . cov(XY,XZ)= Let X be a standard normal random variable. Another random variable is determined as follows. We flip a

3. ### Probability

We have k coins. The probability of Heads is the same for each coin and is the realized value q of a random variable Q that is uniformly distributed on [0,1]. We assume that conditioned on Q=q, all coin tosses are independent. Let

4. ### Math

Alice has two coins. The probability of Heads for the first coin is 1/4, and the probability of Heads for the second is 3/4. Other than this difference, the coins are indistinguishable. Alice chooses one of the coins at random and

1. ### statistics

air coin is flipped 20 times. a. Determine the probability that the coin comes up tails exactly 15 times. b. Find the probability that the coin comes up tails at least 15 times. c. Find the mean and standard deviation for the

2. ### Probability

Alice has two coins. The probability of Heads for the first coin is 1/3, and the probability of Heads for the second is 2/3. Other than this difference, the coins are indistinguishable. Alice chooses one of the coins at random and

3. ### Math

A coin is loaded in such way that a tail is three times as likely to occur as a head. If the coin is flipped twice. Find the probability that two heads occur

4. ### Math

A defective coin minting machine produces coins whose probability of Heads is a random variable Q with PDF fQ(q)={5q4,0,if q∈[0,1],otherwise. A coin produced by this machine is tossed repeatedly, with successive tosses assumed