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or
Var(X) = E[x^2] – μ^2
Var(X+a) = Var(X)
P(ABcCc )+P(AcBCc )+ P(AcBcC)
1 – P(AcBcCc)
(P(ABCc)+P(AcBC)+P(ABcC))
P(both Black) = (b / (a+b)) * (d/(c+d))
If E[X]=5,5 find the value of x2. x2 = 10
P{X<>0} = 1 – P{X=0} = 1 – (5/6)3 = 11/136
E[X|Y=y] = Sum(x*p[X|Y=y] ) = Sum(x*( P[X|Y=y] / P[Y=y]))
E[X|Y=y] = integral(x*f(x|y)) = integral(x* f(XY) / fy(Y))
P{Z<-1.25} = 1- P{Z<1.25} = 0.1056 11%
P(E) = -1
E[X] = 2 and Var(X) = 16/3
Compute the expectation of X. 5/6
Y | -1 0 1 2
P(Y) | 3C 2C 0.4 0.1 The value of the constant C is: 0.1
61. A random variable X has a probability distribution as follows:
X | 0 1 2 3
P(X) | 2k 3k 13k 2k
Then the probability that P(X < 2.0) is equal to 0.25
62. Which one of these variables is a continuous random variable?
The time it takes a randomly selected student to complete an exam.
63. Heights of college women have a distribution that can be approximated by a normal curve with a mean of 65 inches and a standard deviation equal to 3 inches. About what proportion of college women are between 65 and 67 inches tall? 0.25
64. The probability is p = 0.80 that a patient with a certain disease will be successfully treated with a new medical treatment. Suppose that the treatment is used on 40 patients. What is the "expected value" of the number of patients who are successfully treated? 32
65. A medical treatment has a success rate of 0.8. Two patients will be treated with this treatment. Assuming the results are independent for the two patients, what is the probability that neither one of them will be successfully cured? 0.04
66. A set of possible values that a random variable can assume and their associated probabilities of occurrence are referred to as ... Discrete probability distribution
67. Given a normal distribution with μ=100 and σ=10, what is the probability that X>75? 0.9938
68. Which of the following is not a property of a binomial experiment?
There are k possible outcomes on each trial, where k is any positive integer OR
the trials are dependent OR
the probabilities of the two outcomes can change from one trial to the next
69. For a continuous random variable X, the probability density function f(x) represents
the height of the function at x
70. Two events each have probability 0.2 of occurring and are independent. The probability that neither occur is 0.64
71. A smoke-detector system consists of two parts A and B. If smoke occurs then the item A detects it with probability 0.95, the item B detects it with probability 0.98 whereas both of them detect it with probability 0.94. What is the probability that the smoke will not be detected? 0.01
73. A company which produces a particular drug has two factories, A and B. 30% of the drug are made in factory A, 70% in factory B. Suppose that 95% of the drugs produced by the factory A meet specifications while only 75% of the drugs produced by the factory B meet specifications. If I buy the drug, what is the probability that it meets specifications? 0.89 ili 0,81
74. Twelve items are independently sampled from a production line. If the probability any given item is defective is 0.1, the probability of at most two defectives in the sample is closest to … 0.8891
75. A student can solve 6 from a list of 10 problems. For an exam 8 questions are selected at random from the list. What is the probability that the student will solve exactly five problems? 0.53
76. Suppose that 10% of people are left handed. If 8 people are selected at random, what is the probability that exactly 2 of them are left handed? 0.1488
77. Suppose a computer chip manufacturer rejects 15% of the chips produced because they fail presale testing. If you test 4 chips, what is the probability that not all of the chips fail? 0.9995
78. Which of these has a Geometric model? the number of people we survey until we find someone who has taken Statistics
79. In a certain town, 50% of the households own a cellular phone, 40% own a pager, and 20% own both a cellular phone and a pager. The proportion of households that own neither a cellular phone nor a pager is: 30%
80. Four persons are to be selected from a group of 12 people, 7 of whom are women. What is the probability that the first and third selected are women? 7/22
81. Twenty percent of the paintings in a gallery are not originals. A collector buys a painting. He has probability 0.10 of buying a fake for an original but never rejects an original as a fake, What is the (conditional) probability the painting he purchases is an original? 40/41
82. Suppose that the random variable T has the following probability distribution:
t | 0 1 2
P(T = t) | .5 .3 .2
Find . P(t<=0) 0.5
83. A probability function is a rule of correspondence or equation that:
c) Assigns probablities to the various values of x.
84. Which of the following is an example of a discrete random variable?
a. The number of heads obtained when a coin is flipped three times.
b. The number that turns up when a die is rolled.
c. The number of people waiting in line at a movie theater
The number of cows on a cattle ranch.
85. Which of the following is the appropriate definition for the union of two events A and B?
AUB = A+B (occurrence of at least one of event )
86. Johnson taught a music class for 25 students under the age of ten. He randomly chose one of them. What was the probability that the student was under twelve? 1
87. The compact disk Jane bought had 12 songs. The first four were rock music. Tracks number 5 through 12 were ballads. She selected the random function in her CD Player. What is the probability of first listening to a ballad? 7/12
88. Two fair dice, one red and one blue, each have numbers 1-6. If a roll of the two dice totals 6, what is the probability that the red die is showing a 5? 1/5
89. A regular deck of 52 cards contains 4 different suits (Spades, Hearts, Diamonds, and Clubs) that each have 13 cards. If you randomly choose two cards from the deck, what is the probability that both cards will all be hearts? 0.05882 = 5,9%
90. What is the probability of drawing a diamond from a standard deck of 52 cards? ¼ = 25%
91. One card is randomly selected from a shuffled deck of 52 cards and then a die is rolled. Find the probability of obtaining an Ace and rolling an odd number. 1/26
92. The probability that a particular machine breaks down on any day is 0.2 and is independent of the breakdowns on any other day. The machine can break down only once per day. Calculate the probability that the machine breaks down two or more times in ten days. 0.6242
93. Let A, B and C be independent events such that P(A) = 0.5, P(B) = 0.6 and P(C) = 0.1. Calculate P (Ac U Bc U C) 0.73
94. The pdf of a random variable X is given by
What are the values of μ and σ? μ = -1 σ = 2
95. What quantity is given by the formula [Cov(X,Y)]/[sqrt(Var(X)*Var(Y))]
equation for correlation (Cor(X,Y))
96. In the first step, Joe draws a hand of 5 cards from a deck of 52 cards. What is the probability that Joe has exactly one ace? 0.255
97. The number of clients arriving each hour at a given branch of a bank asking for a given service follows a Poisson distribution with parameter λ=3. It is assumed that arrivals at different hours are independent from each other. The probability that in a given hour at most 2 clients arrive at this specific branch of the bank is: 0.42
98. Table shows the cumulative distribution function of a random variable X. Determine P(X>=2)
7/8
99. Table shows the cumulative distribution function of a random variable X. Determine P(X>4) 0
100. Which of the following statements is always true for A and A’?
P(AAc) = 0 P(A+Ac) = 1
101. Consider the universal set U and two events A and B such that…………………. . We know that P(A)=1/3. Find P(B). 2/3
102. A box contains 5 red and 4 white marbles. Two marbles are drawn successively from the box without replacement and it is noted that the second one is white. What is the probability that the first is also white? 3/8
103. If P(A)=1/2 and P(B)=1/2 then P(A U’ B) = 1/4
104. Suppose that P(A|B)=3/5, P(B)=2/7, and P(A)=1/4. Determine P(B|A). 24/35
105. A class contains 8 boys and 7 girls. The teacher selects 3 of the children at random and without replacement. Calculate the probability that the number of boys selected exceeds the number of girls selected. 36/65
106. If the variance of a random variable X is equal to 3, then Var(2X) is : 12
107. Let X and Y be continuous random variables with joint cumulative distribution function
Find P(X>2).
108. Indicate the correct statement related to Poisson random variable
a discrete probability distribution
110. We are given the pmf of two random variables X and Y shown in the tables below. E[X+Y] 5.8
111. The pdf of a random variable X is given by …….. Calculate the parameter a 4
112. Four persons are to be selected from a group of 12 people, 7 of whom are women. What is the probability that three of those selected are women? 35/99
113. Suppose that the random variable T has the following probability distribution:
t | 0 1 2
P(T = t) | .5 .3 .2 Find P(T>=0 and T<2) 0.8
114. Suppose that the random variable T has the following probability distribution:
t | 0 1 2
P(T = t) | .5 .3 .2 Compute the mean of the random variable T. 0.7
115. Three dice are rolled. What is the probability that the points appeared are distinct. 5/9
116. Probability density function of the normal random variable X is given by
What is the standard deviation? σ = 5
117. The event A occurs in each of the independent trials with probability p. Find probability that event A occurs at least once in the 5 trials. 5p(1-p)4
118. The cdf of a random variable X is given by Find the probability P(1.7<X<1.9). 0.4
119. In each of the 20 independent trials the probability of success is 0.2. Find the variance of the number of successes in these trials. 0.25
120. A coin tossed twice. What is the probability that head appears in the both tosses. 1/4