2.2 Let the variable X have the binomial distribution, that is, p(x) = (") p*(q)"-*, x = 0,1,2,-,n If the mean of X is given to be u = np. Find E[X(X – 1)]. ii. Use results in (i) to find the variance of X. i.
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![2.2 Let the variable X have the binomial distribution, that is,
p(x) = (") p*(q)"-*, x= 0,1,2, -- ,n
If the mean of X is given to be µ = np.
Find E[X(X – 1)].
Use results in (i) to find the variance of X.
i.
ii.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32e3e6d3-08c6-47dd-b0fc-968450285203%2F5645bba6-fcd4-4887-8b19-f1d46d639a8a%2Fdhe3pa_processed.jpeg&w=3840&q=75)
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- Suppose X1, ..., Xn have been randomly sampled from a normal distribution with mean 0 and unknown variance sigma^2, and let U = c * i=1 -> n summation (X_ i)^2 , where c is a constant. Find the value of c that minimises the Mean Squared Error (MSE)Suppose you have a sample of size n for variables X and Y. The sample covariance of X and Y is Cov(X,Y) = 1,000, the sample standard deviation for X is Sx=20 and the sample standard deviation for Y is SY=75. What is the sample correlation coefficient, r?Suppose that X has a lognormal distribution with parameters θ = 5 and ω2 = 9. Determine the following: a. P(X < 13,300)b. Value for x such that P(X ≤ x) = 0.95c. Mean and variance of X
- Assume X_1, X_2, .....X_n are random samples from X~Exponential(θ).1) Find the MLE estimator of θ .A random sample of n1 = 16 winter days in Denver gave a sample mean pollution index x1 = 43. Previous studies show that ?1 = 15. For Englewood (a suburb of Denver), a random sample of n2 = 14 winter days gave a sample mean pollution index of x2 = 37. Previous studies show that ?2 = 17. Assume the pollution index is normally distributed in both Englewood and Denver. Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter? Use a 1% level of significance. (a) What is the level of significance? State the null and alternate hypotheses. H0: ?1 < ?2; H1: ?1 = ?2H0: ?1 = ?2; H1: ?1 > ?2 H0: ?1 = ?2; H1: ?1 ≠ ?2H0: ?1 = ?2; H1: ?1 < ?2 (b) What sampling distribution will you use? What assumptions are you making? The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.The standard normal. We assume that both population…