Numerade

Questions asked

INSTANT ANSWER

Suppose that $ f_{X, Y}(x, y)=6(1-x-y) $ for $ x $ and $ y $ defined on the unit squate subject to the restriction that $ 0 \leq x+y \leq 1 $. (a) Find the support of $ X $. Find the support of $ Y $. (b) Sketch the support of $ (X, Y) $ in the xy-plane. (c) Find the marginal pdf, $ f_{X}(x) $, of $ X $. (d) Explain why it is easy to find the marginal pdf of $ Y $ ? (e) Find the MGF, $ M_{X}(t) $, of $ X $. (f) Using $ M_{X}(t) $, find $ \operatorname{var}(X) $. (g) Find $ \operatorname{cov}(X, Y) $. (h) Find the correlation coefficient of $ X $ and $ Y $.

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ANSWERED

Mengchun Cai

Numerade educator

7. A Coca Cola machine has a random supply ( Y ) at the beginning of a given day. It disperses a random amount ( X ) during the day (measured in gallons). The machine is not resupplied during the day. It has been observed that ( X ) and ( Y ) are independent. Assume that ( X ) and ( Y ) have joint pdf: [ f_{X, Y}(x, y)=left{egin{array}{l} frac{1}{2} quad ext { if } 0<x<y, 0<y<2 \ 0 ext { otherwise } end{array} ight. ] Then the amount of Coke in the machine at the end of the day is ( Z=Y-X ). For each of the following, please give a numerical answer when appropriate. (a) Find the marginal pdf, ( f_{X}(x) ), of ( X ). (b) Find the marginal pdf, ( f_{Y}(y) ), of ( Y ). (c) Find ( E[X] ) and ( E[Y] ) (d) Find ( E[Z] ) and ( operatorname{var}(Z) ). (e) Find ( E[X Y] ). (f) Find ( operatorname{cov}(X, Y) ).

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ANSWERED

Lucas Finney

Numerade educator

The number of electric cars sold weekly at Swann’s dealership has mean 16 and variance of 9. (a) Give a lower bound to the probability that next week’s sales are between 10 and 22, inclusively. Use Chebyshev’s inequality. (b) Give an upper bound to the probability that next week’s sales exceed 18 cars. Use Chebyshev’s one-sided inequality.

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Dominador Tan

Numerade educator

Suppose that 𝑓𝑋,𝑌(𝑥, 𝑦) = 6(1 − 𝑥 − 𝑦) for 𝑥 and 𝑦 defined on the unit square subject to the restriction that 0 ≤ 𝑥 + 𝑦 ≤ 1. (a) Find the support of 𝑋. Find the support of 𝑌. (b) Sketch the support of (𝑋, 𝑌) in the xy-plane. (c) Find the marginal pdf, 𝑓𝑋(𝑥), of 𝑋. (d) Explain why it is easy to find the marginal pdf of 𝑌? (e) Find the MGF, 𝑀𝑋(𝑡), of 𝑋. (f) Using 𝑀𝑋(𝑡), find 𝑣𝑎𝑟(𝑋). (g) Find 𝑐𝑜𝑣(𝑋, 𝑌). (h) Find the correlation coefficient of 𝑋 and 𝑌.

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ANSWERED

Hoan Nguyen

Numerade educator

Let 𝑋 be a random variable with MGF, 𝑀(𝑡) = (1 − 𝛽𝑡)^𝛼 where 𝛼 and 𝛽 are positive constants. (a) Find 𝐸[𝑋]. (b) Find 𝑣𝑎𝑟(𝑋). (c) Find 𝐸[𝑋^4].

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James Kiss

Numerade educator

Albertine visits a popular casino in Las Vegas and chooses to play a new game called Simplicity. On each round, she loses $1 with probability 0.7, loses $2 with probability 0.2, or wins $10 with probability 0.1. Albertine returns to her motel after 100 rounds. Approximate the probability that Albertine has lost money as a result of her 100 bets.

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Hoan Nguyen

Numerade educator

A random variable Y has the MGF 𝑀𝑌(𝑡)= 𝑒^(5𝑡) – 4. Calculate the second central moment(variance) of Y.

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Luke Humphrey

Numerade educator

Let X be a random variable with the MGF 𝑀𝑋(𝑡)= 𝑒^2𝑡 – 3𝑒^−𝑡. Find the first four moments (mean, variance, skewness, and kurtosis) of X.

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Hoan Nguyen

Numerade educator

Suppose you are given a continuous random variable U with PDF given by: f(u) = 0.5𝑒^(−𝑢/2) for u ≥ 0. Find the MGF of U.

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ANSWERED

Mengchun Cai

Numerade educator

The MGF of X is given by 𝑀𝑋(𝑡) = exp (2𝑒^𝑡 − 2) and that of Y by 𝑀𝑌 (𝑡) = ((3/4)𝑒^𝑡 +(1/4))^10. If X and Y are independent, find (a) P(X + Y = 2) (b) P(XY = 0) (c) E[XY]

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Pedro C.