GATE Problems In Probability
1GATE Problems in ProbabilityAbstract—These problems have been selected fromGATE question papers and can be used for conductingtutorials in courses related to a first course in probability.1) An urn contains 5 red balls and 5 black balls.Inthe first draw, one ball is picked at randomand discarded without noticing its colour.Theprobability to get a red ball in the second drawis(A)12(B)49(C)59(D)692) There are 3 red socks, 4 green socks and 3 bluesocks.You choose 2 socks.The probability thatthey are of the same colour is(A)15(B)730(C)14(D)4153) The probability that a k-digit number doesNOT contain the digits 0,5, or 9 is(A) 0.3k(B) 0.6k(C) 0.7k(D) 0.9k4) Three fair cubical dice are throen simultaneously. The probability that all three dice havethe same number of dots on the faces showingup is (up to third decimal place).5) Candidates were asked to come to an interview with 3 pens each. Black,blue,green andred were the permitted pen colours that thecandidate could bring. The probability that acandidate comes with all 3 pens having thesame colour is.6) The probability of getting a ”head” in a singletoss of a biased coin is 0.3. The coin is tossedrepeatedly till a ”head” is obtained. If the tossesare independent, then the probability of getting”head” for the first time in the fifth toss is.7) Given Set A [2,3,4,5] and Set B [11,12,13,14,15], two numbers are randomlyselected,one from each set. What is probabilitythat the sum of the two numbers equals 16?(A) 0.20 (B) 0.25 (C) 0.30 (D) 0.338) Consider a dice with the property that theprobability of a face with n dots showing up isproportional to n. The probability of the facewith three dots showing up is.9) Step 1. Flip a coin twice.Step 2. If the outcomes are (TAILS, HEADS)then output Y and stop.Step 3. If the outcomes are either (HEADS,HEADS) or (HEADS, TAILS), then output Nand stop.Step 4. If the outcomes are (TAILS, TAILS),then go to Step 1.The probability that the output of the experiment is Y is (upto two decimal places).10) Let X and Y denote the sets containing 2 and20 distinct objects respectively and F denotethe set of all possible functions defined fromX and Y. Let f be randomly chosen from F.The probability of f being one-to-one is.11) The probability that a given positive integerlying between 1 and 100 (both inclusive) isNOT divisible by 2,3 or 5 is.12) P and Q are considering to apply for a job. The
21probability that P applies for the job is , the4probability that P applies for the job given that1Q applies for the job is , and the probability2that Q applies for the job given that P applies1for the job is . Then the probability that P3does not apply for the job given that Q doesnot apply for the job is(A)45(B)56(C)78(D)111213) Two players, A and B, alternately keep rollinga fair dice. The person to get a six first winsthe game. Given that player A starts the game,the probability that A wins the game is(A)511(B)12(C)713(D)61114) A continuous random variable X has a probability density function f (x) e x , 0 x . Then P (X 1) is(A) 0.368 (B) 0.516) Two independent random variables X and Y areuniformly distributed in the interval [ 1, 1].The1probability that max[X, Y ] is less than is23912(A)(B)(C)(D)4164317) A fair coin is tossed till a head appears for thefirst time. The probability that the number ofrequried tosses is odd,is13(B)12(A)13(B)37(C)12(D)4719) Let X be a random variable with probability 0.2 x 6 1density function f (x) 0.1 1 6 x 6 4 0otherwise.The probability P (0.5 X, 5) is.20) Consider two identically distributed zero-meanrandom variables U and V. Let the cumulativedistribution functions of U and 2V be F(x) andG(x) respectively. Then,for all values of x(A) F (x) G(x) 6 0(C) (F (x) G(x))x 6 0(B) F (x) G(x) 0(D) (F (x) G(x))x 0(C) 0.632 (D) 1.015) A random variable X has probability densityfunction(f (x) as given below:a bx 0 x 1f (x) 0otherwise2If the expected value E[X] , then P r[X 30.5] is.(A)18) A box contains 4 white balls and 3 red balls.In succession, two balls are randomly selectedand removed from the box. Given that the firstremoved ball is white, the probability that thesecond removed ball is red is(C)23(D)3421) Let U and V be two independent and identically distributed random variables such that1P (U 1) P (U 1) . The entropy2H(U V) in bits is(A)34(B) 1(C)32(D) log2 322) Let U and V be two independent zero mean1Gaussian random variables of variances and41respectively. The probability P (3V 2U)9is(A)49(B)12(C)23(D)5923) Two independent random variables X and Yare uniformly distributed in the interval [ 1, 1].1The probability that max [X, Y ] is less than2is
3(A)34(B)916(C)14(D)2324) A binary symmetric channel (BSC) has a tran1sition probability of . If the binary transmit89symbol X is such that P (X 0) , then10the probability of error for an optimum receiverwill be(A)780(B)6380(C)910(D)11025) A fair coin is tossed till a head appears for thefirst time. The probability that the number ofrequried tosses is odd,is(A)13(B)12(C)23(D)3426) A fair dice is tossed two times. The probabilitythat the second toss result in a value that ishigher than the first toss is(A)236(B)26(C)512(D)1227) A fair coin is tossed 10 times. What is theprobability that ONLY the first two tosses willyield heads?(A)(B) 1 2210C2(C) 1 22(D) 1 10210C2(B)116(C)16(B) Both the student and the teacher are wrong(C) The student is wrong but the teacher isright(D) The student is right but the teacher iswrong30) If E denotes expectation, the variance of arandom variable X is given by(A) E[X 2 ] E 2 [X] (C) E[X 2 ](B) E[X 2 ] E 2 [X] (D) E 2 [X]31) An examination consists of two papers, Paper 1and Paper 2. The probability of failing in Paper1 is 0.3 and that in Paper 2 is 0.2. Given thata student has failed in Paper 2, the probabilityof failing in Paper 1 is 0.6. The probability ofa student failing in both the papers is:(A) 0.5 1 10228) Consider two independent random variables Xand Y with identical distributions. The variables X and Y take value 0, 1 and 2 with1 11probabilities ,andrrespectively. What2 44is the conditional probability P (X Y 2 X Y 0)?(A) 0k12345P(X k) 0.1 0.2 0.4 0.2 0.1(A) Both the student and the teacher are right(D) 129) A discrete random variable X takes values from1 to 5 with probabilities as shown in the table.A student calculates the mean of X as 3.5 andher teacher calculates the variance of X as 1.5.Which of the following statements is true?(B) 0.18 (C) 0.12 (D) 0.0632) A probability density function is of the formp(x) Ke α x , x ( , )The value of K is(A) 0.5(B) 1(C) 0.5α (D) α33) Consider a binary digital communication system with equally likely 0’s and 1’s. When binary 0 is transmitted the voltage at the detectorinput can lie between the level -0.25V and 0.25V with equal probability: when binary 1is transmitted, the voltage at the detector canhave any value between 0 and 1V with equalprobability. If the detector has a threshold of2.0V (i.e., if the received signal is greater than0.2V, the bit is taken as 1), the average bit errorprobability is
4(A) 0.15 (B) 0.2(C) 0.05 (D) 0.5(A) pq (1 p)(1 q) (C) p(1 q)(B) pq34) Let X and Y be two statistically independentrandom variables uniformly distributed in therange ( 1, 1) and ( 2, 1) respectively. LetZ X Y , then the probability that [Z 6 2]is(A) zero (B)16(C)13(D)11235) Let X be the Gaussian random variable obtained by sampling the process at t ti andletZ 1 y2 e 2 dyQ(α) 2παThe probability that [x 6 1] is (A) 1 Q(0.5)(C) Q 2 1 2(B) Q(0.5)(D) 1 Q 1 2 2 36) Let Y and Z be the random variables obtainedby sampling X(t) at t 2 and t 4respectively. Let W Y-Z. The variance of Wis(D) 1 pq40) Suppose A and B are two independent eventswith probabilities P (A) 6 0 and P (B) 6 0.e and Be be their complements. Which oneLet Aof the following statements is FALSE?(A) P (A B) P (A)P (B) (C) P (A B) P (A) P (B)(B) P (A B) P (A)e B)e P (A)Pe (B)e(D) P (A41) A digital communication system uses a repetition code for channel encoding/decoding.During transmission, each bit is repeated threetimes(0 is transmitted as 000, and 1 is transmitted as 111). It is assumed that the sourceputs out symbols independently and with equalprobability. The decoder operates as follows: Ina block of three received bits, if the number ofzeros exceeds the number of ones, the decoderdecides in favour of a 0, and if the number ofones exceeds the number of zeros, the decoderdecides in favour of a 1. Assuming a binarysymmetric channel with crossover probabilityp 0.1, the average probability of error is .(A) 13.36 (B) 9.36 (C) 2.64 (D) 8.0037) Let (X1 , X2 ) be independent random variables.X1 has mean 0 and variance 1, while X2 hasmean 1 and variance 4. The mutual informationI (X1 ; X2 ) between X1 and X2 in bits is.38) Let the random variable X represent the number of times a fair coin needs to be tossed tilltwo consecutive heads appear for the first time.The expectation of X is.39) Let X [0, 1] and Y [0, 1] be two independent binary random variables. If P (X 0) p and P (Y 0) q, then P (X Y 1) isequal to42) Two random variables X and Y aredistributedaccording to((x y) 0 6 x 6 10 6 y 6 1fx,y (x, y) 0otherwise.The probability P (X Y 6 1) is .43) A binary communication system makes useof the symbols ”zero” and ”one”. There arechannel errors. Consider the following events: x0 :a ”zero” is transmitted x1 :a ”one” is transmitted y0 :a ”zero” is received y1 :a ”one” is receivedThe following probabilities are given: P (x0 ) 1, P (y0 x0 ) 34 , and P (y0 x1 ) 12 . The2information in bits that you obtain when youlearn which symbol has been received (whileyou know that a ”zero” has been transmitted)is .
5 v 2244) Let X be a zero mean unit variance Gaussianrandom variable. E[ X ] is equal to ., and for v 1, use Q(v) e45) If calls arrive at a telephone exchange such thatthe time of arrival of any call is independentof the time of arrival of earlier or future calls,the probability distribution function of the toatlnumber of calls in a fixed time interval will be(A) 10 7(C) 10 4(B) 10 6(D) 10 2(A) Poisson(C) Exponential(B) Gaussian(D) Gamma46) Consider a communication scheme where thebinary valued signal X satisfiesP {X 1} 0.75 and P {X 1} 0.25.The received signal Y X Z, where Z is aGaussian random variable with zero meanand variance σ 2 . The received signal Y is fedto the threshold detector. The output of thethreshold detectorX̂ is:( 1 Y τX̂ 1 Y 6 τTo achieve minimum probability of errorP {X̂ 6 X}, the threshols τ should be(A) strictly positive (D) strictlypositive,zero or strictly negative depending on(B) zerothe nonzero valueof σ 2(C) strictly negative47) Consider a discrete-time channel Y X Z,where the additive noise Z is signal-dependent.In particular, given the transmitted symbolX { a, a} at any instant, the noisesample Z is chosen independently from aGaussian distribution with mean βX and unitvariance. Assume a threshold detector withzero threshold at the receiver.When β 0, the BER was found to beQ(a) 1 10 8. Z u21e 2 duQ(v) 2π v When β 0.3, the BER is closet to48) Consider the random processX(t) U V t,where U is a zero-meaan Gaussian randomvariable and V is a random variable distributedbetween 0 and 2. Assume that U and V arestatistically independent. The mean value of therandom process at t 2 is.49) Consider the Z-channel given in Fig. 1. Theinput is 0 or 1 with equal probability.Fig. 1.If the output is 0, the probability that the inputis also 0 equals.50) If P and Q are two random events, then thefollowing is TRUE:(A) Independence of P and Q implies thatPr P Q 0(B) Pr(P Q) Pr(P ) Pr(Q)(C) If P and Q are mutually exclusive, thenthey must be independent(D) Pr(P Q) 6 Pr(P )
651) A fair coin is tossed three times in succession.If the first toss produces a head, then theprobability of getting exactly two heads inthree tosses is:(A)18(C)38(B)12(D)3452) The probability density function (PDF) of arandom variable X is as shown in Fig. 2.Fig. 3.Fig. 2.The corresponding cumulative distributionfunction (CDF) has the form(A) Fig. 3(C) Fig. 4Fig. 4.(B) Fig. 4(D) Fig. 653) The input X to the binary Symmetric Channel(BSC) shown in the figures is ’1’ withprobability 0.8. The cross-over probability is1. If the received bit Y 0, the conditional7probability that ’1’ was transmitted is.54) The distribution function fx (x) of a randomvariable X is shown in Fig. 8. The probabilitythat X 1 is(A) Zero(C) 0.55(B) 0.25(D) 0.3055) Probability density function p(x) of a randomvariable x is as shown below. The value of α is
7Fig. 5.Fig. 6.(A)2c(B)1c
8For testing the null hypothesis H0 : f f0against the alternative hypothesis H1 : f f1at level of significance α 0.19, the power ofthe most powerful test isFig. 7.(A) 0.729(C) 0.615(B) 0.271(D) 0.38557) Let the probability density function of a random variable X be 0 x 21 xf (x) c(2x 1)2 12 x 1 0otherwise.Then,the value of c is equal toFig. 8.58) Suppose X and Y are two random variablessuch that aX bY is a normal random variable for all a, b R. Consider the followingstatements P,Q,R and S:(P): X is a standard normal random variable.(Q): The conditional distribution of X givenY is normal.(R): The conditional distribution of X givenX Y is normal.(S): X - Y has mean 0.Which of the above statements ALWAYS holdTRUE?Fig. 9.(C)2(b c)(D)1(b c)56) Let X be a random variable with probabilitydensity functionf {f0 , f1 }, where(2x 0 x 1f0 (x) 0otherwise(3x2 0 x 1f1 (x) 0otherwise(A) both P and Q(C) both Q and S(B) both Q and R(D) both P and S59) Let X be a random variable with the followingcumulative distribution function:0 x 0 x2 0 x 12F (x) 31 x 1 2 41 x 1. ThenP 41 X 1isequalto
960) Let X1 be an exponential random variablewith mean 1 and X2 a gamma random variablewith mean 2 and variance 2. If X1 and X2 areindependently distributed,then P (X1 X2 ) isequal toCommon Data for the next two Questions :Let X and Y be jointly distributed randomvariables such that the conditional distributionof Y , given X x, is uniform on the interval(x 1, x 1). Suppose E(X) 1 andV ar(X) 35 .61) The mean of the random variable Y is(A)12(C)32bution function: 0 1 4F (x) 13 1 21Then E(X) is equal to65) Let X and Y be two random variables havingthe joint probability density function(2 0 x y 1f (x, y) 0 otherwise.Thentheprobability conditional32isequaltoP X 3 Y 4(A)(B) 1(D) 2(B)1223(C) 1(D) 263) Let the random variable Xfunction: 0 x 2F (x) 35 1 x8 2 1ThenP (2 X 4)have the distributionx 00 x 11 x 22 x 3x 3.isequal5923(B)(C)79(D)8966) Let Ω (0, 1] be the sample space and let P (·)be a probability function defined by(x0 x 12P ((0, x]) 2x 12 x 1. Then P { 21 } is equal to62) The variance of the random variable Y is(A)x 00 x 11 x 22 x 113.x 113to64) Let X be a random variable having the distri-67) Suppose the random variable U has uniformdistribution on [0, 1] and X 2 log U. Thedensity of X(ise x x 0(A) f (x) 0otherwise.(B) f (x) (2e 2x0x 0otherwise.(C) f (x) (1 x2e2x 0otherwise.(D) f (x) (0120x [0, 2]otherwise.
1068) Suppose X is a real-valued random variable.Which of the following values CANNOTbe attained by E[X] and E[X 2 ], respectively?1213(A) 0 and 1(C)(B) 2 and 3(D) 2 and 5andLet X and Y be random variables having thejoining probabilitydensity function 11(x y)2 2y x , 2πy ef (x, y) 0 y 1 0otherwise72) The variance of the random variable X is69) Let Xn denote the sum of points obtained whenn fair dice are rolled together. The expectationand variance of Xn are(A)(B)35 27n andn respectively. (C)212357n andn respectively.212 n n735andrespec212tively.(B)1213(C)(D)141671) Consider two identical boxes B1 and B2 , wherethe box B(i 1, 2) contains i 2 red and5 i 1 white balls. A fair die is cast. Let thenumber of dots shown on the top face of the diebe N. If N is even or 5, then two balls are drawnwith replacement from the box B1 , otherwise,two balls are drawn with replacement from thebox B2 . The probability that the two drawnballs are of different colours is7(A)25(B)925112(C)712(B)14(D)512(D) None of the above70) Let X and Y be jointly distributed randomvariables having the joint probability densityfunction (1x2 y 2 6 1πf (x, y) 0 otherwiseThen P (Y max(X, X)) (A)(A)12(C)25(D)1625Common Data for the next two Questions :73) The covariance between the random variablesX and Y(A)13(C)16(B)14(D)112Common Data for the next two Questions :Let X and Y be continuous random variableswith the joint probability density function(ae 2y 0 x y f (x, y) 0otherwise74) The value of a is(A) 4(C) 1(B) 2(D) 0.575) The value of E(X Y 2) is(A) 4(C) 2(B) 3(D) 1
1176) Let X and Y be two random variables havingthe joint probability density function(2 0 x y 1f (x, y) 0 otherwiseThen the conditional probability P (X 62 Y 34 ) is equal to3(A)(B)59(C)23(D)798977) Let Ω (0, 1] be the sample space and let P (.)be a probability( function defined byx0 6 x 122P ((0, x]) x 1 6x61 2 Then P { 12 } is equal to.78) Let X be a random variable with the followingcumulative distribution function: 0 x 0 x2 0 6 x 12F (x) 31 6x 1 2 41 x 1ThenP ( 14 x 1) is equal to.79) Let X1 be an exponential random variable withmean 1 and X2 a gamma random variable withmean 2 and variance 2. If X1 and X2 areindependently distributed, then P (X1 X2 )is equal to.Common Data for the next two Questions :Let X and Y be two continuous random variables with (the joint probability density function2 0 x y 1, x 0, y 0f (x, y) 0 elsewhere.80) P (X Y 12 ) is(A)14(B)12(C)34(D) 181) E(X Y 12 )(A)14(B)12(C) 1(D) 282) If a random variable X assumes only positiveintegral values, with the probabilityP (X x) 32 ( 13 )x 1 , x 1, 2, 3, .,then E(X) is(A)292(B)3(C) 1(D)3283) The joint probability density function of tworandom variablesX and Y is given as 6 (x y 2 ) 0 6 x 6 10 6 x 6 1f (x, y) 5 0elsewhereE(X) and E(Y ) are, d5584) Suppose the random variable U has uniformdistribution on [0, 1] and X 2 log U. Thedensity of X is(e x x 0(A) f (x) 0otherwise
12(2e 2x(B) f (x) 0 1 xe(C) f (x) 2 2 0 1(D) f (x) 2 0x 0otherwisex 0otherwisex [0, 2]otherwise85) Suppose X is a real-valued random variable.Which of the following values CANNOT beattained by E[X] and E[X 2 ], respectively?11and23(A) 0 and 1(C)(B) 2 and 3(D) 2 and 5
GATE Problems in Probability Abstract—These problems have been selected from GATE question papers and can be used for conducting tutorials in courses related to a first course in probability. 1) An urn contains 5 red balls and 5 b
NOTE: In a DUAL GATE INSTALLATION the gate opener on the same side of the driveway as the control box is known as the MASTER GATE OPENER and that gate is refered to as the MASTER GATE. Conversly the gate opener on the other gate is refered to as the SLAVE GATE OPENER and the gate is refered to as the SLAVE GATE. For Mighty Mule FM702, GTO/PRO .
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Gate Drain Gate controls this. Gate can not control below that. So current can leak through there. PDSOI Gate 1V Gate controls this. No leakage path. FDSOI Gate 1V Leak Source Drain FinFET Si Gate 1V Gate Source Drain Better Electrostatics Stronger Gate Control - Lower V t for the same leakage - Shorter channel for the same V t
This GATE study material can be downloaded as PDF so that your GATE preparation is made easy and you can ace your exam. These study notes are important for GATE EC, GATE EE, GATE ME, GATE CE and GATE CS. They are also important for IES, BARC, BSNL, DRDO and the rest. Before you get sta
Pros and cons Option A: - 80% probability of cure - 2% probability of serious adverse event . Option B: - 90% probability of cure - 5% probability of serious adverse event . Option C: - 98% probability of cure - 1% probability of treatment-related death - 1% probability of minor adverse event . 5
the unit size output resistance, the unit size gate area capacitance and the gate perimeter capacitance of gate i , respectively. Although the gate shown here is a two-input AND gate, the model can be easily generalized for any gate with any number of input pins. Fig. 4. The model of component i , which is a wire segment, by a -type RC circuit.
No. Per API 650 Section 3.2.4, the tank is limited to one in. of water vacuum (roughly 25 mm). If the tank must be designed for 50 mm water vacuum, then this is a special design which is not covered by API 650. 3.2.4 9th - May 1993 Should the vacuum relief system set pressure be 25 mm of water if the required design vacuum condition is 50 mm of water? API 650 does not cover the required .
