7. Suppose the proportion X of surface area in a randomly selectedquadrate that is covered by a certain plant has a standard beta distributionwith a = 6 and ß = 2.a) Compute E(X) and V(X).b) Compute P(X < 0.4).c) Compute P(0.1l

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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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Let X and Y have a bivariate normal distribution with μ = 70, X Find (a) E(Y|X= 76) (b) Var(Y | X= 76) (c) P (Y≤ 861X=76) 0²: X = 100, = 80, My o² = 169, and p=5/13.
Let X,...,X6 be a random sample of size 6 from Normal(0, o2). a. What is the constant "a" so that a (X1-X3) Y = 2x+ $x+x+ x is t-distribution? b. Compute P(Y < 0.95).
1. Let y1,.. , Yn be a random sample from a uniform distribution on the interval (0, B). Let B1 2 Li=1 Yi and B, = ) Max(y1,.., Yn) be estimators for B. Find the sampling distribution of B2 Show that B, and B, are unbiased estimators for B. 11.
A machine generates rectangular tiles with randomly varying length (L) and width (W). Let the length follow L ~Gamma(100,10) distribution and the width follow W ~Gamma(50,10) distribution, and define the area of the tile as S = L × W, and its perimeter as Q = 2(L+W). For parts (a-c) below, assume the length and width are independent (i.e. LI W). (a) Show that the expected value of the area is E[S] = 50. (b) Show that the variance of the area is Var[S] = 75.5. (c) Use Chebyshev's inequality to find an upper bound on P(S > 75). For parts (d,e) below, assume the length and width are correlated with Corr(L, W) = -0.5. (d) Show that the expected value of the area is E[S] ~ 49.64645. (e) Find the mean and variance of the perimeter: E[Q] and Var[Q].
5. The distribution of male weight in the United States has a standard deviation of roughly o = 25 lbs. The average weight u varies from state to state. Suppose you randomly select 100 males in Louisiana and ask their weights, and you compute the average weight Xn of the n = 100 males. a) Assuming that the standard deviation of weight in Louisiana is 25 lbs, what does the Central Limit Theorem tell you the (approximate) distribution of X will be?
1a) Use the normal approximation to find the indicated probability. The sample size is n, the population proportion of successes is p, and X is the number of successes in the sample.n = 100, p = 0.32: P(X < 34) b) A mail-order company has a 9% success rate. If it mails advertisements to 500 people, find the probability of getting fewer than 38 sales. c) Find the critical value zα/2 needed to construct a(n) 99.2% confidence interval. Please answer all parts of the question 1a to 1c. Thanks.
The pressure of air (in MPa) entering a compressor is measured to be X = 8.5 ± 0.2, and the pressure of the air leaving the compressor is measured to be Y = 21.2 ± 0.3. The intermediate pressure is therefore measured to be P = √XY = 13.42 . Assume that X and Y come from normal populations and are unbiased. a) From what distributions is it appropriate to simulate values X* and Y*? b) Generate simulated samples of values X*, Y*, and P*. c) Use the simulated sample to estimate the standard deviation of P. d) Construct a normal probability plot for the values P*. Is it reasonable to assume that P is approximately normally distributed? e) If appropriate, use the normal curve to find a 95% confidence interval for the intermediate pressure.
5. Y1, Y2,..., Yn are iid normal (u, o²) where o² unknown (п-1)52 Show that is a pivot quantity for estimation of o?
b) Let X₁, X₂,..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25 functions Mx₁(t) = ³t+t² and My(t) = (₁-¹)²5, respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5. ii) What is the value of the sample size n, if P[Σ1(X¡ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - µy| ≥10) < 0.04?
The random variable x has a normal distribution with mean 50 and variance 9. Find the value of x, call it x0, such that: a) P(x ≤ xo) = 0.8413 b) P(x > xo) = 0.025 c) P(x > xo) = 0.95 d) P(41 ≤ x ≤ xo) = 0.8630
4. Let X denote the mean of a random sample of size n from the distribution N(μ, 10) and P(x - 1²/ < μ< * + ²) = O A A. 176 C. 20 = 0.964, then the value n of is equal to B B. 13
b) Find the mean and standard deviation of the difference. c) Find the rejection region and state your decision at α = 0.01.

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7. Suppose the proportion X of surface area in a randomly selected quadrate that is covered by a certain plant has a standard beta distribution with a = 6 and ß = 2. a) Compute E(X) and V(X). b) Compute P(X < 0.4). c) Compute P(0.1l