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In Problem 8 t, suppose that the white balls are numbered, and let
a.
b.
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- Given the probability mass function of a random variable X p(x) = c/(x+1) if x = 0,1 and p(x) = 0 in other mass points. What is the value of c ?arrow_forwardQuestion 1 : Suppose that the probability density function (p.d.f.) of the life (in weeks) of a certain part is f(x) = 3 x 2 (400)3 , 0 ≤ x < 400. (a) Compute the probability the a certain part will fail in less than 200 weeks. (b) Compute the mean lifetime of a part and the standard deviation of the lifetime of a part. (c) To decrease the probability in part (a), four independent parts are placed in parallel. So all must fail, if the system fails. Let Y = max{X1, X2, X3, X4} denote the lifetime of such a system, where Xi denotes the lifetime of the ith component. Show that fY (y) = 12 y 11 (400)12 , y > 0. Hint : First construct FY (y) = P(Y ≤ y), by noticing that {Y ≤ y} = {X1 ≤ y} ∩ {X2 ≤ y} ∩ {X3 ≤ y} ∩ {X4 ≤ y}. (d) Determine P(Y ≤ 200) and compare it to the answer in part (a)arrow_forward4 (b) An insurance company provides customers with both auto and home insurance policies. For a particular customer, Χ is the deduction on his or her auto policy and Y is the deduction on the home policy. Possible values of Χ are K100 and K250, and for Y are K0, K100 and K200. The joint probability density function for ( ) ,YΧ is given by the following table: Χ Y K100 K250 K0 0.20 0.05 K100 0.10 0.15 K200 0.20 0.30 iv. If we look only at those insurance customers selecting the lowest auto mobile insurance deduction (K100), what is the probability that a randomly selected\ customer will also select the lowest home deduction (K0). v. Compute the correlation coefficient of Χ and Yarrow_forward
- Suppose that the random variables X,Y, and Z have the joint probability density function f(x,y,z) = 8xyz for 0<x<1, 0<y<1, and 0<z<1. Determine P(X<0.7).arrow_forwardLet p(x) = cx^2 for the integers 1, 2, and 3 and 0 otherwise. What value must c be in order for p(x) to be a legitimate probability mass function?arrow_forwardA survey of cars on a certain stretch of highway during morning commute hours showed that 70% had only one occupant, 15% had 2, 10% had 3, 3% had 4, and 2% had 5. Let X represent the number of occupants in a randomly chosen car. a) Find the probability mass function of X. b) Find P(X ≤ 2). c) Find P(X > 3). d) Find μX. e) Find σX.arrow_forward
- If the probability mass function of the variable X described in the table is. find variance Y=x^2+4x If you know that the torque is of the second order of the variable X about the origin is equal to 2.85arrow_forwardQUESTION 10 Suppose f(x) = 1/4 over the range a ≤ x ≤ b, and suppose P(X > 4) = 1/2. What are the values for a and b? a. 2 and 6 b. Cannot answer with the information given. c. 0 and 4 d. Can be any range of x values whose length (b − a) equals 4. QUESTION 11 The probability density function, f(x), for any continuous random variable X, represents: a. all possible values that X will assume within some interval a ≤ x ≤ b. b. the probability that X takes on a specific value x. c. the height of the density function at x. d. None of these choices. QUESTION 12 Which of the following is true about f(x) when X has a uniform distribution over the interval [a, b]? a. The values of f(x) are different for various values of the random variable X. b. f(x) equals one for each possible value of X. c. f(x) equals one divided by the length of the interval from a to b.…arrow_forwardThe General Social Survey asked a sample of adults how many siblings (brothers and sisters) they had (X) and also how many children they had (Y). We show results for those who had no more than 4 children and no more than 4 siblings. Assume that the joint probability mass function is given in the following contingency table: y x 0 1 2 3 4 0 0.03 0.01 0.02 0.01 0.01 1 0.09 0.05 0.08 0.03 0.01 2 0.09 0.05 0.07 0.04 0.02 3 0.06 0.04 0.07 0.04 0.02 4 0.04 0.03 0.04 0.03 0.02 Find ρ(X, Y). (Round the final answer to four decimal places.) ρ(X, Y) =arrow_forward
- The General Social Survey asked a sample of adults how many siblings (brothers and sisters) they had (X) and also how many children they had (Y). We show results for those who had no more than 4 children and no more than 4 siblings. Assume that the joint probability mass function is given in the following contingency table: y x 0 1 2 3 4 0 0.03 0.01 0.02 0.01 0.01 1 0.09 0.05 0.08 0.03 0.01 2 0.09 0.05 0.07 0.03 0.02 3 0.06 0.04 0.07 0.04 0.02 4 0.04 0.04 0.04 0.03 0.02 Find the conditional expectation E(Y|X = 4). (Round the final answer to four decimal places.)arrow_forwardThe General Social Survey asked a sample of adults how many siblings (brothers and sisters) they had (X) and also how many children they had (Y). We show results for those who had no more than 4 children and no more than 4 siblings. Assume that the joint probability mass function is given in the following contingency table: y x 0 1 2 3 4 0 0.03 0.01 0.02 0.01 0.01 1 0.09 0.05 0.08 0.03 0.01 2 0.09 0.05 0.07 0.03 0.02 3 0.06 0.04 0.07 0.04 0.02 4 0.04 0.04 0.04 0.03 0.02 Find the conditional probability mass function pY|X(y|4). (Round the final answer to four decimal places.) pY|X(0|4) = pY|X(1|4) = pY|X(2|4) = pY|X(3|4) = pY|X(4|4) = Find the conditional probability mass function pX|Y (x|3). (Round the final answer to four decimal places.) pX|Y(0|3) = pX|Y(1|3) = pX|Y(2|3) = pX|Y(3|3) = pX|Y(4|3) = Find the conditional expectation E(X|Y = 3). (Round the final answer to four decimal places.) E(X|Y = 3) =arrow_forwardRework problem 16 in section 4.2 of your text, involving drawing markers from a box of markers with ink and markers without ink. Assume that the box contains 12 markers: 9 that contain ink and 3 that do not contain ink. A sample of 6 markers is selected and a random variable Y is defined as the number of markers selected which do not have ink. Find the probability density function. Be certain to list the values of Y in ascending order.arrow_forward
- A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON