The sample space of a random experiment is the set of positive real numbers, S={x|x>0}. Define events A and B as A={x | x > 40} and B= {x|x < 65). Describe each of the following events: a) B' = b) ANB
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- The sample space of a random experiment is the set of positive real numbers, $={x 1 x >0} Define events A and B as A={x | x > 40} and B={x | x < 65). Describe each of the following events: a) A' b) AUB5.) Suppose that a coin is tossed twice so that the sample space is S = {HH, HT, TH, TT}. Let X represent the # of tails that can come up. Create a table for its Probability Distribution Function (PDF) and draw its histogram. Test if the sum of probability is 1.2. In statistics, the mean x and variance S of a sequence of numbers X1, X2, X3,' are defined as: Xi (x; – x)² n i=1 п i=1 and use the above definition to prove that: a) ΣT1지-치 = 0 b) s² = (÷E%1x;?)– x²
- b) Let X₁, X₂,..., X5 be a random sample of size 5 from the standard normally population. If the statistic Y is given by Y = X1 X2 X3 √√X₁²+X3²+X5²² i) Briefly explain why Y follows a Student's t distribution with 3 degrees of freedom. ii) Compute the probability that Y is between 2.35 and 5.84. iii) What is the mean and variance of the statistic Y? QUESTION OThe sample space of a random experiment is the set of positive real numbers, S={x | x>0}. Define events A and B as A={x | x > 40} and B={x | x < 65}. Describe each of the following events: a) B' b) ANBAge Age Gender Under 40 40 or Older Total Male 12 2 14 Female 8 3 11 Total 20 5 25 The table above shows the distribution of age and gender for 25 people who entered a contest. If the contest winner will be selected at random, what is the probability that the winner will be either a female or 40 or older? 13/25 16/25 3/25 10/25
- The sample space of a random experiment is the set of positive real numbers, $={x 1 x >0} Define events A and B as A={x | x > 40} and B={x | x < 65). Describe each of the following events: a) A' b) AUBThere are two boxes. In each box there are 4 cards with a different number printed on it. The four cards have been numbered as 1,2,3,4 in each box. Two cards are drawn random from each box. The random variable X represents the difference between the number on the card from box 1 minus the number on the card from box 2.a) Find the mean of this probability distribution. (i.e. Find E[X] )b) Find the variance and standard deviation of this probability distribution.(i.e. Find V[X] and SD[X])The random variables M and W are defined as follows:M = X+5 and W = (1/2)X+5c) Find E[M] and E[W].d) Find V[M] and V[W].e) Mary and William play a game using the cards in the above boxes. Randomly two cards are drawn, and Mary records his score using the random variable M and William uses the random variable W. They repeat this for a large number of times and compare their scores. Comment on any likely differences or similarities of their scores.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?
- Determine the distribution of the random variable X and sketch its graph. for a E (-00, -1) x € [-1, }) for e € [},2) 品 for for a) F(x) = x € [2, ) æ € },+) 17 20 for too) 1The sample space of a random experiment is the set of positive real numbers, $={x 1 x >0} Define events A and B as A={x | x > 40} and B={x | x < 65). Describe each of the following events: a) A' b) AUBAssume that A, B, and C are subsets of a sample space S with Pr(A) = 0.8, Pr(B) = 0.6, Pr(C) = 0.15. 1. Find Pr(A'), Pr(B'), and Pr(C'). Pr(A') = Pr(B') = Pr(C') = 2. If Pr(A U B) = 0.85., find Pr(AN B). Pr(AN B) = 3. Suppose that we know that B and C are disjoint events. Find Pr[BUC. Pr[BUC] = 4. Suppose that instead of being disjoint C C B. Find Pr[B U C]. Pr[BUC] =