Problems1. Consider discrete random variables X and Y [BLAK 1979] with the joint pmf asshown below:Y0# # ##Are X and Y independent? Are they uncorrelated?124

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Answered: Problems 1. Consider discrete random…

A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Problems 1. Consider discrete random variables X and Y [BLAK 1979] with the joint pmf as shown below: Y X -1 0 1 1 -2 16 -1 1 2 곪 16 16 Are X and Y independent? Are they uncorrelated? -100 100-19 -19-19-19-19 -100 100-1
6. A Bernoulli trial can be modeledd as a random variable X that can take on only two values: X=0 and X= 1, which are murnally exclusive. If P(X 1) 3 p. what is (a) PCX = 0) ? G Var(N) Earropy of N
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The answer is 23
3. Students looked at the effect of a certain fertilizer on plant growth. The students tested this fertilizer on one group of plants (Group A) and did not give fertilizer to a second group (Group B). Let X and Y denote the respective growths of the plants (in mm) in Group A and Group B over six weeks. Suppose X and Y are independent random variables with distributions N(ux, o) and N(uy, o), respectively. A random sample from N(ux.o%) of size n = 25 yielded i 35.83 and s = 23.81, while a random %3! sample from N(uy o) of size m = 29 yielded y = 31.51 and s = 33.76. %3D (a) Assume o = o, test the null hypothesis at 1% significance level that the mean growths are equal against the alternative that the fertilizer enhanced growth. (b) If o #o, test the mull hypothesis at 1% significance level that the mean growths are equal against the alternative that the fertilizer enhanced growth. (c) Test Ho : o = oỷ against H1 : of + oỷ at the a = 0.05 significance level.
Problems 4.3. Suppose that X and Y are jointly discrete random variables with for x = 1, 2, . n and 2 p(x, y) = {n(n + 1) y = 1, 2, . otherwise Compute Px(x) and py(y) and determine whether X and Y are independent.
1) a. A random variable X has the following probability distribution: X 0 × B 5 x B 10 x B 15 х В 20 x B 25 x B P(X = x) 0.1 2n 0.2 0.1 0.04 0.07 а. Find the value of n. Find the mean/expected value E(x), variance V(x) and standard deviation of the given probability distribution. Find E(-4A x + 3) and V(6B x – 7) b. С.
Question 7: Let X be a random variable such that E(X – 2)² = 10 and E(X+1)² = 4, then а. Е(X) b. Е(X) %3D с. Е (X) %3D d. E(X) =
Problem 7 Let X ~ Poisson(a) and Y ~ Poisson(3) be two independent random variables. Define a new random variable as Z = X +Y. Find the PMF of Z.
7.15 Suppose that X₁, X₂, ..., X and Y₁, Y₂,..., Y are independent random samples, with the variables X, normally distributed with mean, and variance of and the variables Y, normally distributed with mean 2 and variance a2. The difference between the sample means, X - Y, is then a linear combination of m + n normally distributed random variables and, by Theorem 6.3, is itself normally distributed. a Find E(X-Y). b Find V(X-Y). c Suppose that of = 2, o₂ = 2.5, and m = n. Find the sample sizes so that (X-Y) will be within 1 unit of (₁-₂) with probability .95.

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Problems 1. Consider discrete random variables X and Y [BLAK 1979] with the joint pmf as shown below: Y 0 1 # # # 1 2 13 Are X and Y independent? Are they uncorrelated?