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- The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.If np > 5 and nq > 5 estimate P(more than 4) with n = 13 and p = 0.5 by using the normal distrubution as an approximation to the binomial distribution. P(at least 4) =Show proof and provide derivations -A binomial distribution is symmetric at p=1/2 and approaches symmetry when n → ∞ for a fixed p
- Let X have a Poisson distribution with parameter λ. Show that E(X)=λ directly from the definition of expected value. (Hint: The first term in the sum equals 0, and then x can be canceled. Now factor out λ and show that what is left sums to 1.)Suppose n = 1,000. Use the Central Limit Theorem, which tells us that P (hat) is approximately normally distributed, to provide an approximate value for the probability that the difference between P (hat) and P exceeds 0.05. Your value may depend on PLet X be an exponential random variable with standard deviation σ. FindP(|X − E(X)| > kσ ) for k = 2, 3, 4, and compare the results to the boundsfrom Chebyshev’s inequality.
- Show that the following belong to the exponential family1. The pareto distribution: f(y; θ) = θy^−θ−1 2. The Negative Binomial distribution:If np(is greater than or equal to)5, estimate P(fewer than 5) with n=13 and p=0.4 by using the normal distribution as an approximation to the binomial distribution; if np<5 or nq<5, then state that the normal approximation is not suitable. P(fewer than 5)=?If np≥5 and nq≥5, estimate P(more than 7) with n=11 and p=0.3by using the normal distribution as an approximation to the binomial distribution; if np<5or nq<5, then state that the normal approximation is not suitable. P(more than 7)=
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