Numerade

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Danielle Fairburn

Numerade educator

17. A random variable ( X ) has a CDF such that [ F(x)=left{egin{array}{ll} x / 2 & 0<x leqslant 1 \ x-1 / 2 & 1<x leqslant 3 / 2 end{array} ight. ] (a) Graph ( F(x) ). (b) Graph the pdf ( f(x) ). (c) Find ( P[X leqslant 1 / 2] ). (d) Find ( P[X geqslant 1 / 2] ). (e) Find ( P[X leqslant 1.25] ). (f) What is ( P[X=1.25] ) ?

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Danielle Fairburn

Numerade educator

A continuous random variable X has pdf given by f(x) = c(1 - x)x² if 0 < x < 1 and zero otherwise. (a) Find the constant c. (b) Find E(X).

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Supratim Pal

Numerade educator

7. A discrete random variable X has a pdf of the form f(x) = c(8 - x) for x = 0, 1, 2, 3, 4, 5, and zero otherwise. (a) Find the constant c. (b) Find the CDF, F(x). (c) Find P[X > 2]. (d) Find E(X).

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Supratim Pal

Numerade educator

In a bolt factory, machines 1, 2, and 3 respectively produce 20%, 30%, and 50% of the total output. Of their respective outputs, 5%, 3%, and 2% are defective. A bolt is selected at random. (a) What is the probability that it is defective? (b) Given that it is defective, what is the probability that it was made by machine 1?

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Daniel Carr

Numerade educator

25. A box contains three good cards and two bad (penalty) cards. Player A chooses a card and then player B chooses a card. Compute the following probabilities: (a) P(A good). (b) P(B good | A good). (c) P(B good | A bad). (d) P(B good ? A good) using (1.5.5). (e) Write out the sample space of ordered pairs and compute P(B good ? A good) and P(B good | A good) directly from definitions. (Note: Assume that the cards are distinct.) (f) P(B good). (g) P(A good | B good).

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Eduard Sanchez

Numerade educator

Let P(A) = 1/2, P(B) = 1/8, and P(C) = 1/4, where A, B, and C are mutually exclusive. Find the following: (a) P(A ? B ? C). (b) P(A' ? B' ? C').

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Ivan Kochetkov

Numerade educator

19. Let P(A) = P(B) = 1/3 and P(A ? B) = 1/10. Find the following: (a) P(B'). (b) P(A ? B'). (c) P(B ? A'). (d) P(A' ? B').

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Hoan Nguyen

Numerade educator

3 (3 points) Find the joint MGF of the continuous random variables X and Y with joint pdf f(x, y) = e^-y if 0 < x < y < ? and zero otherwise.

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Ivan Kochetkov

Numerade educator

Let X_1, X_2, ..., X_n be a random sample of size n from a normal distribution, X_i ~ N(u, sigma^2), and let X~_n be the sample median. Find constants m and c such that X~_n is asymptotically normal N(m, c^2/n).

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INSTANT ANSWER

12. A certain type of weapon has probability \( p \) of working successfully. We test \( n \) weapons, and the stock pile is replaced if the number of failures, \( X \), is at least one. How large must \( n \) be to have \( P[X \geqslant 1] \doteq 0.99 \) when \( p=0.95 \) ? (a) Use exact binomial. (b) Use normal approximation. (c) Use Poisson approximation. (d) Rework (a) through (c) with \( p=0.90 \).

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Alexis