Your pocket contains a random number N of coins, where N follows the Poisson distribution with parameter d. All the coins are identical and weighted so that the probability of landing on head is p. Call X the random variable giving the number of heads obtained after tossing all the coins. Show that X is distributed as a Poisson random variable with parameter Xp.
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- X is an poisson random variable with parameter λ = 4 Calculate P {X ≤ 3}Let X be a Poisson random variable with parameter λ = 5.Compute the probabilities p0, . . . . , p6 to four decimal places.According to an agreement between a smartphone producer and a retail company, the former delivers a large batch of smartphones of some given model. The retail company applies a quality check by randomly selecting 100 smartphones from the batch, and, if 3 or more of these smartphones are found defected, the entire batch is rejected. Suppose a historical defect rate for this type of smartphones is p = 0.005. Derive (and compute) the probability that the batch is not rejected using an approximation with the Poisson distribution.
- A company has 8000 arrivals of Internet traffic over a period of 17,460 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= μx•e−μ x! to find the probability of exactly 3 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?A company has 9000 arrivals of Internet traffic over a period of 20,740 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)=μx•e−μx! to find the probability of exactly 3 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?A company has 9000 arrivals of Internet traffic over a period of 18,050 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= (μ^x • e^−μ) / x! to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?