Recall the Geometric(p) distribution where X = number of flips of a coin until you get...
Recall the Geometric(p) distribution where X = number of flips of a coin until you get a head (H) with Pr(H) = p. The distribution is Pr(X = x) = (1 − p) (x−1) p for x = 1, 2, . . . , with mean E(X) = ∑ x=1∞ (x(1 − p) (x−1) p) = 1/p, which can be obtained by brute force. An easier way to find the mean is to condition on the first toss, say Y = 0 or 1 if the first toss is T or H. Show the mean is 1/p using E(X) = EE(X | Y).
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Recall the Geometric(p) distribution where X = number of flips
of a coin until you get...
Problem 0.2 Recall the Geometric(p) distribution where X-number of flips of a coin until you get...
Problem 0.2 Recall the Geometric(p) distribution where X-number of flips of a coin until you get a head (H) with Pr(H) -p. The distribution is Pr(X- (1-p)1p for 1,2,. , with mean E(X)x(1 - p)*-p- 1/p, which can be obtained by brute force. An easier way to find the mean is to condition on the first toss, say Y- 0or 1 if the first toss is T or H. Show the mean is 1/p using E(X) EE(X Y)
Problem 0.1 Let X be the number of people who enter a bank by time t>0....
Problem 0.1 Let X be the number of people who enter a bank by time t>0. Suppose ke-t k! for k 0,1,2,., and for t>s > 0, and k-r=0,1,2, . . . . (a) Find Pr(X2 = k | X,-1) for k = 0, 1, 2, . . . . (b) Find E[X2 X1-1 Useful information: Don't eat yellow snow, and et-L=0 tk/k! Problem 0.2 Recall the Geometric(p) distribution where Xnumber of flips of a coin until you get a...
Problem o.1 Let X, be the number of people who enter a bank by time t...
Problem o.1 Let X, be the number of people who enter a bank by time t > 0. Suppose k! for k- 0,1,2,..., and s (t - s)k-e-t for t>s> 0, and k2r 0,1,2,.... (a) Find Pr[X2 k| X 1 for k 0,1,2,.... (b) Find E2 X1 1 Useful information: Don't eat yellow snow, andeot/k! Problem o.2 Recall the Geometric(p) distribution where X- number of flips of a coin until you get a head (H) with Pr(H) - p. The...
Q3. Suppose we toss a coin until we see a heads, and let X be the...
Q3. Suppose we toss a coin until we see a heads, and let X be the number of tosses. Recall that this is what we called the geometric distribution. Assume that it is a fair coin (equal probability of heads and tails). What is the p.m.f. of X? (I.e., for an integer i, what is P(X=i)? What is ?[X]? ({} this is a discrete variable that takes infinitely many values.)
I have an unfair coin for which P(H) = p, where 0 < p < 1....
I have an unfair coin for which P(H) = p, where 0 < p < 1. I toss the coin repeatedly until I observe heads for the first time. Let Y, be the total number of coin tosses. Find the distribution of Y. Hint: the range of Y is {1,2,3...}.
The geometric distribution is a probability distribution of the number X of Bernoulli trials needed to...
The geometric distribution is a probability distribution of the number X of Bernoulli trials needed to get one success. For example, how many attempts does a basketball player need to get a goal. Given the probability of success in a single trial is p, the probability that the xth trial is the first success is: Pr(x = x|p) = (1 - p*-'p for x=1,2,3,.... Suppose, you observe n basketball players trying to score and record the number of attempts required...
A coin with probability p is tossed until the first head occurs. It is then tossed...
A coin with probability p is tossed until the first head occurs. It is then tossed again until the first tail occurs. Let X be the total number of tosses required. 1) Find the distribution function of X. 2) Find the mean and variance of X.
You flip a coin 100 times. Let X= the number of heads in 100 flips. Assume...
You flip a coin 100 times. Let X= the number of heads in 100 flips. Assume we don’t know the probability, p, the coin lands on heads (we don’t know its a fair coin). So, let Y be distributed uniformly on the interval [0,1]. Assume the value of Y = the probability that the coin lands on heads. So, we are given Y is uniformly distributed on [0,1] and X given Y=p is binomially distributed on (100,p). Find E(X) and...
If random variable X counts the number of coin flips to get a Tail, then it...
If random variable X counts the number of coin flips to get a Tail, then it has the following PDF x 1 2 3 4 ... k f ( x ) 2/3 2/9 2/27 2/81 ... 2/3^k What is P ( x ≥ 4 ) Answer choices: 8/27 2/27 26/27 None of these 1/27
A coin with probability p of heads is tossed until the first head occurs. It is...
A coin with probability p of heads is tossed until the first head occurs. It is then tossed again until the first tail occurs. Let X be the total number of tosses required. (i) Find the distribution function of X. (ii) Find the mean and variance of X
Problem 0.2 Recall the Geometric(p) distribution where X-number of flips of a coin until you get a head (H) with Pr(H) -p. The distribution is Pr(X- (1-p)1p for 1,2,. , with mean E(X)x(1 - p)*-p- 1/p, which can be obtained by brute force. An easier way to find the mean is to condition on the first toss, say Y- 0or 1 if the first toss is T or H. Show the mean is 1/p using E(X) EE(X Y)
Problem 0.1 Let X be the number of people who enter a bank by time t>0. Suppose ke-t k! for k 0,1,2,., and for t>s > 0, and k-r=0,1,2, . . . . (a) Find Pr(X2 = k | X,-1) for k = 0, 1, 2, . . . . (b) Find E[X2 X1-1 Useful information: Don't eat yellow snow, and et-L=0 tk/k! Problem 0.2 Recall the Geometric(p) distribution where Xnumber of flips of a coin until you get a...
Problem o.1 Let X, be the number of people who enter a bank by time t > 0. Suppose k! for k- 0,1,2,..., and s (t - s)k-e-t for t>s> 0, and k2r 0,1,2,.... (a) Find Pr[X2 k| X 1 for k 0,1,2,.... (b) Find E2 X1 1 Useful information: Don't eat yellow snow, andeot/k! Problem o.2 Recall the Geometric(p) distribution where X- number of flips of a coin until you get a head (H) with Pr(H) - p. The...
The geometric distribution is a probability distribution of the number X of Bernoulli trials needed to get one success. For example, how many attempts does a basketball player need to get a goal. Given the probability of success in a single trial is p, the probability that the xth trial is the first success is: Pr(x = x|p) = (1 - p*-'p for x=1,2,3,.... Suppose, you observe n basketball players trying to score and record the number of attempts required...
A coin with probability p of heads is tossed until the first head occurs. It is then tossed again until the first tail occurs. Let X be the total number of tosses required. (i) Find the distribution function of X. (ii) Find the mean and variance of X
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