1. Consider two random variables X and Y with joint PMF given in the tablea. Find P(X = 2, Y<2)b. Find marginal PMF of X and Yc. Find P(Y=2 |X= 1)Are X and Y independent1231/60.250.12520.1251/61/62. Let X and Y be jointly continuous random variables with joint PDFfw(Xy) =х+ су?0,OSXS1,0

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For a random sample X1, X2, ..., X, following an Exponential(rate = 1/μ) distribution, we have seen that the large sample distribution of the sample mean X follows a Normal distribution with mean parameter µ and the variance parameter μ²/n. Suppose that we have a sample of size n = 25 from an exponential population with mean μ and we want to test n Hoμ = 2 v/s Hд μ # 2. Use the large sample distribution of ☑ to compute an approximate p-value for the test if we observe x = 2.6 and state the conclusion of your test at the 5% level of significance.
4. Question 4: Y is a discrete random variable with PMF S1/3, y=1 p(y) = 2/3, y = 2 (a) Find the MGF for Y. (b) Develop the first two raw moments from MGF and find the variance V(Y). Answers:
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3.7. Consider the performance function Y = 3x1-2x2 where Xi and X2 are both normally distributed random variables with Ax' = 16.6 0% 2.45 μΧ2 = 18.8 ơx.-2.83 The two variables are correlated, and the covariance is equal to 2.0. Determine the probability of failure if failure is defined as the state when Y 0 3.8. The resistance (or capacity) R of a member is to be modeled using R = R,MPF where Rn is the nominal value of the capacity determined using code procedures and M, P, and Fare random variables that account for various uncertainties in the capacity. If M, P, and F are all lognormal random variables, determine the mean and variance of R in terms of the means and variances of M, P, and F.
Test the claim about the difference between two population means mu 1μ1 and mu 2μ2 at the level of significance alphaα. Assume the samples are random and independent, and the populations are normally distributed. Claim: mu 1μ1equals=mu 2μ2; alphaαequals=0.01 Population statistics: sigma 1σ1equals=3.3, sigma 2σ2equals=1.5 Sample statistics: x overbar 1x1equals=17, n1equals=29, x overbar 2x2equals=19, n2equals=28 Determine the alternative hypothesis. Upper H Subscript aHa: mu 1μ1 ▼ not equals≠ greater than or equals≥ less than or equals≤ greater than> less than< mu 2μ2 Determine the standardized test statistic. zequals=nothing (Round to two decimal places as needed.) Determine the P-value. P-valueequals=nothing (Round to three decimal places as needed.) What is the proper decision? A. Reject Upper H 0H0. There is enough evidence at the 1% level of significance to reject the claim. B. Fail to reject Upper H 0H0. There is not…
X1,.Xn and Y1,.„Yn are two random samples selected from independent normal distributions with variances Var(X)=s2 and Var(Y)=2s2, respectively o2>0. 2U+V U = E-(X –- X)² ve V = En (Y, – Y)? has been defined . According to this information U = 202 Which of the following is the distribution of the random variable M? O A. t(2n-2) O B. x² (2(n-1)) O C. t(2n-1) O D. x² (2n-1)
25. Suppose X is a random variable distributed with mean u and standard deviation a > 0. If the random variable Y is defined as Y = u(X - uX)/a, then E(Y) is: a. 0 b. 1 d. E(X)
Suppose you have a normally distributed population such that o? = 7. Using the sampling distribution of S?, we find that from a random sample of size n = 11, s² = 16.053. Does our population variance seem || reasonable? Note: We will consider our population variance reasonable if our x value falls within the interval x0.975 (df), x'o.025 (df)]. 1) Find the values of the interval [xo.975(df), x²o.025(df)]. x°0.975(df) = (Round to 3 decimals.) x'0.025(df) - (Round to 3 decimals.) 2) Find the value of x that comes from the data you collected in the problem statement. (Round to 3 decimals.) 3) Is the population variance reasonable? Yes O No
Suppose we obtain n observations from a sinusoid contaminated by Gaussian noise: X = A· sin(2 · T· f·k+¢) + Wk, k = 1, 2, ...,n, %3D where Wg are independent identically distributed (iid) Gaussian random variables with mean o and variance o?. The frequency f and phase o are known constants, while A is an unknown constant. Find the maximum likelihood estimator for A. Show all work.
Match the following.
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1. Consider two random variables X and Y with joint PMF given in the table Find P(X = 2, Y <2) Find marginal PMF of X and Y Find P(Y=2 |X= 1) Are X and Y independent a. b. 3 С. 1 1/6 0.25 0.125 d. 2 0.125 1/6 1/6