Let X be a discrete random variable with probability mass function in the form 5 4 3 2 1 X k 3k 2k 3k k (fx k is
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Q: An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of "heads" (h) and "tails" (t)…
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- Let Y be a discrete random variable with generating function 4 Gy (s) 6 - s What is E(Y) (in decimal)? Answer:1. Suppose a discrete random variable Xhas probability function given by 5. 10 12 0.13 POX-1) 0.08 0.16 025 16 2y 0.03 005 Solve for y andfind the mean and variance of X.The probability generating function for a random variable X is given by: 1 1,25 1 9x(z) =z2 + 1 -z20 710 + 2 P(X < 5 |X < 13) equals Select one: а. 0.8 b. 0.2 C. 1 d. 0.286 е. О
- B5. Let X₁, X₂, ..., Xn be IID random variable with common expectation µ and common variance o², and let X = (X₁ + + X₂)/n be the mean of these random variables. We will be considering the random variable S² given by (a) By writing or otherwise, show that S² (b) Hence or otherwise, show that n S² = (x₁ - x)². = Ĺ(X₂ i=1 X₁ X = (X₁-μ) - (x-μ) = Σ(X; -μ)² - n(X - μ)². i=1 ES² = (n-1)0². You may use facts about X from the notes provided you state them clearly. (You may find it helpful to recognise some expectations as definitional formulas for variances, where appropriate.) (c) At the beginning of this module, we defined the sample variance of the values x₁, x2,...,xn to be S = 1 n-1 n i=1 ((x₁ - x)². Explain one reason why we might consider it appropriate to use 1/(n-1) as the factor at the beginning of this expression, rather than simply 1/n. B6. (New) Roughly how many times should I toss a coin for there to be a 95% chance that between 49% and 510/ of my nain toon land Honda?Suppose that the genders of the three children of a certain family are soon to be revealed. Outcomes are thus triples of "girls" (g) and "boys" (b), which we write gbg, bbb, etc. For each outcome, let R be the random variable counting the number of girls in each outcome. For example, if the outcome is bbg, then R(bbg) = 1. Suppose that the random variable X is defined in terms of R as follows: X=R- - 2R-4. The values of X are given in the table below. Outcome bbb ggb bbg gbg gbb bgg bgb gg Value of X-4 -4 -5 -4 -5 -4 -5 -1 Calculate the values of the probability distribution function of X, i.e. the function py. First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row. Value x of X Px (x) E 1:48 PM 3/21/2022 hp Compag LAI956X 立An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of "heads" (h) and "tails" (t) which we write hth, ttt, etc. For each outcome, let N be the random variable counting the number of tails in each outcome. For example, if the outcome is tth, then N (tth)=2. Suppose that the random variable X is defined in terms of N as follows: X=N²-2N-2. The values of X are given in the table below. Outcome ttt htt hhh tht tth hth hht thh Value of X 1 -2 -2 -2 -2 -3 -3 -3 Calculate the probabilities P (X=x) of the probability distribution of X. First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row. Value X of X P(X=x) 0 0 0 0 0 00 X Ś
- Let X and Y be two discrete random variables with the joint probability distribution is given by: Y= f(x.y) 2 1/6 1/3 1/12 X= 1/6 1/12 2 0 1/6 Then E ([(X + 2Y)®]) is equal to: None of these 147/12 57/6 281/12The probability generating function for a random variable X is given by: 9x(z) = 1 1 10. 1 I 20 8 -25 + 4 P(X > 10 |X < 22) equals Select one: a. 0.286 b. 0 c. 0.8 d. 1 е. 0.2Suppose a and b be two possible values of a random variable X with a > b. The probability that X lies between a and b is P(a > X > b) = F (a) - F (b) Select one: O True O False
- An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of "heads" (h) and "tails" (t) which we write hth, ttt, etc. For each outcome, let N be the random variable counting the number of tails in each outcome. For example, if the outcome is hth, then N (hth) = 1. Suppose that the random variable X is defined in terms of N as follows: X=2N² − 6N-1. The values of X are given in the table below. Outcome thh tth hhh hth ttt htt hht tht Value of X-5 -5 -1 -5 -1 -5 -5 -5 Calculate the probabilities P(X=x) of the probability distribution of X. First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row. Value X of X P(X=x) 0 0 00 XWhat is the definition of independence for two discrete random variables X and Y? X and Y are independent if and only if P(X= x) = P(Y= y) for all x and y. OX and Y are independent if and only if P(X = x Y = y) = P(X = x)*P(Y= y) for all x and y. X and Y are independent if and only if P(X= x | Y = y) = P(X = x) for all x and y. PreviousEach of the random variables X and Y takes only 3 values {1,2,3} with the following probabilities: 1 1 0 1/6 1/6 y 2 1/6 0 1/6 3 1/6 1/6 0 2.