Let X be a random variable with p.d.f f(x) =)",x = 1,2,3,. Find P(A), where A={1,3,5, ..} 2/3 2/5 O 4/5 3/5
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- 2. Let X, X,.., x, be a random sample from a distribution with p.d.f. f(x;0)=0x, |>x>0 Is the MLE O EE of 0? AEE of 0?Let the following simple random sample X1, X2, · · , X11 following: 1. Binomial pmf. (11, ¾); 2. Uniform pmf; 3. Uniform pdf (0, a); 4. Exponential pdf with (µ). Find the corresponding pmf/pdf of Y1 , Y4, Y7 and F(Y;) where Yi < Y½ < .. < Y6 < ..< Yı1.Suppose the random variable X follows a normal distribution with mean 80 and variance 100. Then the probability is 0.6826 that X is in the symmetric interval about the mean between two numbers are A) -2 to 2 B) -3 to 3 C) -1 to 1 D) none
- B4. Let X₁,... Xn ~ N(μ, o2) be independent random variables. 2 (a) From lectures we know (X=X) ²³. X₂ (b) Let n i=1 ~ x²(v). What is the value of v? n 1 *³ =, ²-₁, [(x₁ - x) ². Σ(x s n 1 i=1 Determine Var(s), that is, the variance of the sample variance. (c) Assume now that we have observed data ₁,...,n ER with sample variance s². Use the result from part (a) to find numbers an and b, such that [ans, bns] is an exact 95%-confidence interval for o².Let Y1,..., Y, be i.i.d. random variables from the Weibull distribution f(y; 0) Σ (멤) exp ), y > 0. Show that ô = Li=1í is an unbiased estimator of 0. Is Ô an efficient estimaotr of 0?Prove the following
- Q.3. Compute the probability that: (a) their sum is odd; (b) their product is even. Three distinct integers are chosen at random from the first 15 positive integers.Q 6.1. Suppose Z = (Z1, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i ≤ 3, Zi~ N(0, 1) are i.i.d. random variables. Each of the random variables, (a)-(d), on the left is equal in distribution to exactly one random variables, (1)-(4), on the right. Pair up according to "equal in distribution" and explain briefly your reasoning. (a) (X₂) (¹) (X) = (2) 1/√2 (Ⓒ) (x₂) - (3/1² ¹1/1²) (2) (c = 2 0 = 2 Z₁ 1) (²) 3 (4¹) (X) = (²¹) (1) (2) (3) Y₁ 1 (29) - ()) (2) = Y₂ 1 Y₁ (121) = (1/² √₂) (2) (1/√2 1/√2 /2 −1/√√2, Y₂ X₁ X₂ = 22 1 1 2 0 Y₁ 2 1 (4) (2)) = (²) (²) 1 1 Z₁ Z₂ Z3True False? a)A sided dice with the probability of 1 and 6 coming p and the other numbers (2, 3, 4, 5) is 2p is rolled n times. x̄, is the average of the numbers in n shots. According to this information, p = 2 / 5x̄. b)If X1, ..., Xn, N is a random sample drawn from the distribution (μ, ϭ²), the sample mean x̄ and sample variance S² are always independent.
