5. Let the random variable Y,, have a distribution that is b(n, p). Prove that (Y/n)(1-Yn/n). converges in probability to p(1-p).
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- 1. Let X ~ Poisson(A) and Y ~ Poisson(u). Assume that X and Y are independent. Use probability generating functions to find the distribu- tion of X + Y.Suppose that the number of traffic accidents, N, in a given period of time is distributed as a Poisson random variable with E(N)=100. Use the normal approximation to the Poisson to find A such that P(100 - AQ1) 14% of the adults in a certain population are infected by a Corona Virus. Five adults are selected randomly from this population for diagnoses. Find the probability that at least two of them will be infected by .this Virus 1 .Q2) Suppose that f(X)=X,12). a) Find .b) Find the mean Q3) A study shows that the systolic blood pressures for adults in a certain population are approximately normally distributed with mean of 115 and .standard deviation of 10 a) Find the probability that a randomly selected person will have a blood -pressure between 109 and 124 Sel).b) Find the reading that is exceeded by only 5% of the population التي يتجاوزها 5% من المجتمع(5 Both K and M has standard normal distribution and they are independent random variables. Z=K+M Find E[K(K^2+M)^(1/4)].A company has 6000 arrivals of Internet traffic over a period of 12,160 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 μ*• e -μ to find the probability of X! Internet arrivals have a Poisson distribution. If we want to use the formula P(x) = 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? μ= (Round to three decimal places as needed.) X = e = (Round to three decimal places as needed.) tv -O). ☐☐ + S il 1 √₁ Vi A O (LI) More W Next XAssume that X is a random variable whose conditional distribution given the variable Y is poisson P (X | Y) = Po (Y). Suppose further that Y has a gamma distribution Y ∼ Gamma (1, 1). (a) Determine the value E (XY). (b) Determine the conditional distribution P (Y | X).Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?Assume that n independent count variables {X,,X2,...,X,} are identically distributed as X, ~ Poi(0) where i= 1,2,...,n and to estimate E(X,)= 0, consider the sample mean estimator ô = -5x,. n (b) Use the expectation of the distribution of S = nô to compute the expectation E(@) and the bias of this estimator. State whether this estimator unbiased. (c) Use the variance of the distribution of S= nô to examine var(Ô) and the standard error of this estimator. Use the variance and the bias of ê to compute the Mean Squared Error (MSE) of this estimator. Comment on the MSE behaviour in the asymptotic limit as n→o. State whether this estimator is consistent or (d) not.Calculate P(X>2)c) Suppose that a random variable Y is uniformly distributed on an interval (0,1) and let c> 0 be a constant. i) Find the moment generating function of X = -cY. ii) What is the distribution of X? 2| Page5.7. Let the random variable Y, have a distribution that is b(n, p). (a) Prove that Y,/n converges in probability to p. This result is one form of the weak law of large numbers. (b) Prove that 1 Y/n converges in probability to 1-p.
