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131A_1_Scan100305115008

# 131A_1_Scan100305115008 - 216 Chapter4 One RandomVariable...

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Unformatted text preview: 216 Chapter4 One RandomVariable Problems 217 (b) Show the mapping from S to \$2. (c) Find the region in the square corresponding to the event {Z s z}. ((1) Find and plot the cdf of Z. (e) Use the cdftoli11d2P[Z > 01, P[Z > b1, P[Z 519/21, PlZ > 17/41. . , 2 4.8. Let g be a point selected at random from the unit interval. Consrder the random 1 16 O*O/ W X z (] ,,_ §>7l/2. I y ' l (a) Sketch X as a function of g”. _'1 t (I) 1. , x (b) Find and plot the edl of X. “'(é) Find the probability of the events {X > 1}, {5 < X < 7}, {X s 20}. 4.9. The loose hand 01' a clock is spun hard and the outcome é’ is the angle in the range where the hand comes to rest. Consider the random variable ; = 2 sing/4). (a) Sketch X as a function of g. (b) Find and plot the cdf of X. (c) Find the probability of the events {X > 1'}, {—1/2 < X < 1/2}. {X s 1/‘ 4.10. Repeat Problem 4.9 if 80% of the time the hand comes to rest anywhere in the cire 20% of the time the hand comes to rest at 3. 6. 9, or 12 o’clock. - HGUREP41 (3) Plot the cdf of onr B = 0.5, 1, and 2. (b) Find the probability 1’le < X < (j + DA] and P[X > j/\1. (c) Plot log PlX > x] vs. log x. I 6. The random variable Xhas cdf: 4.11. The random variable X is uniformly distributed in the interval [#1, 21. 0 x < O (it) Find and plot the cdf orX. Fth) 2 0.5 + csin2(7Tx/2) o g x S 1 (b) Use the cd[ to find the probabilities of the following events: {X l x > 1 {lX — 0.5l < 1},andC = {X > —U.5}. 4.12. The cdf of the random variable X is given by: (a) What values can C assume? (b) Plot the cdf. (c) FindPlX > 0], A L m o 0 x 0.5 -1 S x F X : _ . Xl ) (1 + 10/2 S x g 1 {rectlon 4.2: The Probability Density Function 1 x .>_ 1 4.17. A random variable Xhas pdl': (a) Plot the cdf and identify the type 01' random variable. (b) Find P[X S ~11,P[X : —1].P[X < 0.5],Pl—05 < X < 0.51,P[X >_ [’[X s 21.1’[X > 31. - 4.13. A random variable X has cdf: O elsewhere. thx) : {ca ‘ XZ) ‘1 S x S 1 (a) Find 6 and plot the pdf. (b) Plot the cdf of X. _ (c) Find P[X : 01,P[0 < X < 0.5], and PHX — 0.5t < 0.25“ .18. A random variable X has pdf: 1 l. fX(X) I {mu- 7 x2) 0 S x 5’1 0 elsewhere. 0 for x < 0 FXU‘) = 1 A— 1:62" l'orx 2 0. (3) Plot the cdf and identify the type of random variable. (b) FindPlX S 2],P[X : (J1.P[X < 01, P172 < X < 6],P[X > 101. (4:14.. The random variable X has cdl shown in Fig. P41 1 X (a) What type of random variable is X? (b) Find the following probabilities: P[X < «1],P1X S —11.P[—1 < X <- P[~0.5 s X < 01.11505 3 X s 0.51.PHX — 0.5t < 0.51. 4.15. For B > 0 and )t > 0, the Weibull random variable Yhas cdf: ((1a)): Find 6 and plot the pdf. (b) Plot the cdf of X. (c) Find Flt) < X < 0.5],P[X : i], P[.25 < X < 0.5]. 111 Pro] )16m 4 6 ﬁnd and 101 th K . , 6 d[ f [11 V . I ' . p . ‘ p 0 e I rlndOm Vallable It thC dlstallce 1.101“ the _ 0 (b) Lise the pdf to find the probability that the dart is outside the bull’s eye. . (E) Find and plot the pdf of the random variable Z in Problem 4.7. ( ) Use the pdf to find the probability that the minimum is greater than b/3. 0 for x < 0 FXlx) : {1 — [ix/A)” for x Z 0. ...
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