QUESTION 1 A certain binomial experiment consists of 100 independent Bernoulli t
ID: 3359182 • Letter: Q
Question
QUESTION 1
A certain binomial experiment consists of 100 independent Bernoulli trials with the probability of success is 0.55. X is the random variable which counts the successes. Determine the following probabilities:
0.1346, 0.8654, 0.8151
0.1346, 0.8172, 0.8638
0.1346, 0.8172, 0.8151
0.1827, 0.8172, 0.8151
0.75 points
QUESTION 2
The density function of a continuous random variable X is given by
Find the expected value of X.
E[X] = x
E[X] = 1.5
E[X] = 0
E[X] = 1
0.75 points
QUESTION 3
A six-sided fair die is rolled once. Let X be the random variable that assigns the value of the outcome (i.e., X(w) = 1 means that the outcome was a 1, etc.) What is the expected value, E(X)?
1
7/2
3
1/6
0.75 points
QUESTION 4
Which of the following statements are not correct?
The study of continuous random variables requires the continuous mathematics of the calculus - integrals and derivatives.
In general, each outcome of an experiment can be associated with a number by specifying a rule of association.
The number of movies you watched last year is an example of a continuous random variable.
To study basic properties of discrete random variables, only the tools of discrete mathematics -summation and differences - are required.
0.75 points
QUESTION 5
A fair coin is tossed 5 times. What is the probability of a run of 3 Heads (i.e., an outcome contains 3 Heads in succession, but not 4 or 5).
1/32
5/32
10/32
3/32
0.75 points
QUESTION 6
The lifetime of certain type of light bulbs is exponentially distributed with an average of 1000 hours. What is the probability that a given light bulb will last more 900 hours?
0.5934
0.6321
0.4066
0.3679
0.75 points
QUESTION 7
A random variable X has gamma distribution with parameters alpha = 2 and beta = 3. Find P(6< x < 12 ).
0.017
0.500
0.983
0.314
0.75 points
QUESTION 8
A random variable X has gamma distribution with parameters alpha = 2 and beta = 3. Find P(x < 6 ).
0.577
0.801
0.594
0.983
0.75 points
QUESTION 9
A random variable X has gamma distribution with parameters alpha = 2 and beta = 3. Find P(x > 9).
0.677
0.199
0.801
0.323
0.75 points
QUESTION 10
Which of the following is (are) conditions of a binomial experiment?
There is a sequence of n trials, where n is fixed in advance of the experiment.
The trials are identical, and each trial can result in one of the same two possible outcomes, which we denote by success (S) or failure (F).
The trials are independent, so that the outcome of any particular trial does not influence the outcome of any other trial.
The probability of success is the same (constant) from trial to trial; we denote this probability by p.
All of the above are conditions of a binomial experiment.
0.75 points
QUESTION 11
The probability density function of a random variable X is given by f(x) = 1 for 0 <= x <= 1 and 0 elsewhere. Then P(X = 0.25) =
0
0.00001
0.5
0.25
0.75 points
QUESTION 12
A random number generator in a CAS produces pseudo-random numbers which are uniformly distributed in any given interval. Suppose that the generator is asked to produce 10,000 numbers between 20 and 25. About how many numbers do you expect to get greater than 24? (Assume continuity).
1000
2500
200
2000
0.75 points
QUESTION 13
The number of gallons of gasoline sold daily by a gas station is approximately uniformly distributed between 4000 and 4200 gallons. The profit in dollars is given by the function f(x) = 0.45x - 480 where x is the number of gallons sold. What is the expected profit per day?
$1845
$4100
$480
$1365
0.75 points
QUESTION 14
Let X be a discrete random variable with probability distribution
The variance and the standard deviation of X are
V(X) = 1.02, SD(X) = 1.05
V(X) = -1.05, SD(X) = 1.02
V(X) = 1.05, SD(X) = 1.02
V(X) = -1.02, SD(X) = -1.05
0.75 points
QUESTION 15
Let X be a discrete random variable with E(X) = 0 and V(X) = 1. Find E(20X + 100) and V(20X + 100).
E(20X + 100) = 100, V(20X + 100) = 500
E(20X + 100) = 100, V(20X + 100) = 400
E(20X + 100) = 0, V(20X + 100) = 400
E(20X + 100) = 0, V(20X + 100) = 0
0.75 points
QUESTION 16
Forty percent of the students taking Statistics 101 this semester bought a new book while the remaining 60% either bought a used book already had one. If four students taking Statistics 101 are selected at random, what is the probability that exactly two bought new books?
0.50
0.5248
0.3456
0.8208
0.75 points
QUESTION 17
A company sells three different types of 2 GB USB's at different prices depending on the make, type, and other characteristics. Brand A sells for $12.00, brand B for $25.00, and brand C for $40.00. Of these USB's sold, 65% are brand A, 25% are brand B, and 10% are brand C. The profit for the company is 40% of the retail value, so Y = 0.4X is the rv for the profit. Find the expected value and the variance for the profit.
E(Y) = $4.74, V(Y) = $22.47
E(Y) = $11.86, V(Y) = $269.23
E(Y) = 11.86, V(Y) = $6.55
E(Y) =$7.22, V(Y) = $13.45
0.75 points
QUESTION 18
The function p assumes the following values:
What value must p assume at x = 3 for p to be a probability distribution?
Any positive value.
Any value between 0 and 1.
0
0.12
0.75 points
QUESTION 19
A random variable X has the following cumulative distribution function:
0 for x < 0
0.2 for 0 <= x < 1
F(x) = { 0.5 for 1 <= x < 2
0.9 for 2 <= x < 4
1 for 4 <= x
The probability distribution of X is
0.75 points
QUESTION 20
A random variable X has binomial distribution with parameters n = 75 and p = 0.75. Find the expected value and the variance of X.
E(X) = 18.75, V(X) = 3.75
E(X) = 18.75, V(X) = 14.0625
E(X) = 56.25, V(X) = 14.0625
E(X) = 56.25, V(X) = 3.75
a.0.1346, 0.8654, 0.8151
b.0.1346, 0.8172, 0.8638
c.0.1346, 0.8172, 0.8151
d.0.1827, 0.8172, 0.8151
Explanation / Answer
Ans:
3)
E(X)=1*(1/6)+2*(1/6)+3*(1/6)+4*(1/6)+5*(1/6)+6*(1/6)=(1/6)*(1+2+3+4+5+6)=21/6=7/2
Option b is correct.
5)
P(x=3)=5C3*(1/2)3*(1/2)2=10*(1/32)=10/32
Option C is correct.
6)
P(X>900)=e-(900/1000)=e-0.9=0.4066
Option C is correct.
10)
Option E is correct.(all conditions are required for binomial distribution)
20)
E(X)=75*0.75=56.25
Var(X)=75*0.75*0.25=14.0625
Option C is correct.
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