- •Indicate the formula of computing variance of a random variable X with expectation µ.
- •Indicate the expectation of a Poisson random variable X with parameter .
- •Indicate the variance of a Poisson random variable X with parameter .
- •Indicate the formula for conditional expectation.
- •If a fair die is tossed twice, the probability that the first toss will be a number less than 4 and the second toss will be greater than 4 is
- •If one person is selected randomly, the probability that it did not pass given that it is female is:
- •If X and y are independent random variables with ,,and,,,. Thenis
- •If p(e) is the probability that an event will occur, which of the followings must be false?
- •If one person is selected randomly, what is the probability that it did not pass given that it is male.
- •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?
- •If the variance of a random variable X is equal to 3, then Var(3x) is :
- •Indicate the correct statement related to Poisson random variable .
- •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.
- •Indicate the pdf for standard normal random variable.
- •Indicate the function that can be cdf of some random variable.
- •Indicate the function that can be pdf of some random variable.
- •If two random variables X and y have the joint density function, , find the conditional pdf.
- •If two random variables X and y have the joint density function, , find the conditional pdf.
The probabilities that three men hit a target are respectively 1/6, 1/4 and 1/3. Each man shoots once at the target. What is the probability that exactly one of them hits the target?
11/72
21/72
31/72
3/4
17/72
A problem in mathematics is given to three students whose chances of solving it are 1/3, 1/4, 1/5. What is the probability that the problem will be solved?
0.2
0.8
0.4
0.6
1
You are given P[A∪B] = 0.7 and P[A∪Bc] = 0.9 . Determine P[A] .
0.2
0.3
0.4
0.6
0.8
An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is 0.44 . Calculate the number of blue balls in the second urn.
4
20
24
44
64
The probability that a boy will not pass an examination is 3/5 and that a girl will not pass is 4/5. Calculate the probability that at least one of them passes the examination.
11/25
13/25
1/2
7/25
16/25
A bag contains 5 red discs and 4 blue discs. If 3 discs are drawn from the bag without replacement, find the probability that all three are blue.
1/21
2/21
1/7
4/21
1/3
Find the variance for the given probability distribution.
X |
0 |
2 |
4 |
6 |
P(x) |
0.05 |
0.17 |
0.43 |
0.35 |
1.5636
2.8544
1.6942
2.4484
1.7222
A bag contains 5 white, 7 red and 8 black balls. Four balls are drawn one by one with replacement, what is the probability that at least one is white?
0.7182
Формулой Бернулли называется формула
Indicate the formula of computing variance of a random variable X with expectation µ.
How would it change the variance of a random variable X if we add a number to the X?
Var(X+a)=Var(X)+a
Var(X+a)=Var(X)+a2
Var(X+a)=Var(X)
Var(X+a)=Var(X)
Var(X+a)=Var(X)+a2
How would it change the expected value of a random variable X if we multiply the X by a number k.
Which of the following expressions indicates the occurrence of exactly one of the events A, B, C?
Which of the following expressions indicates the occurrence of at least one of the events A, B, C?
Which of the following expressions indicates the occurrence of all three events A, B, C simultaneously?
Which of the following expressions indicates the occurrence of exactly two of events A, B, C?
Conditional probability P(A|B) can be defined by
Urn I contains a white and b black balls, whereas urn II contains c white and d black balls. If a ball is randomly selected from each urn, what is the probability that the balls will be both black?
The table below shows the probability mass function of a random variable X.
xi |
0 |
x2 |
5 |
pi |
0.1 |
0.2 |
0.7 |
If E[X]=5,5 find the value of x2.
3
1
12
0.8
10
The probability of machine failure in one working day is equal to 0.01. What is the probability that the machine will work without failure for 5 days in a row.
0.99999
0.95099
1
0.05
0.55
The cumulative distribution function of a discrete random variable X is given by Find P{3<X<9}.
0,4
0,5
0,6
0,9
1
A fair die is rolled three times. A random variable X denotes the number of occurrences of 6’s. What is the probability that X will have the value which is not equal to 0.
91/216
125/216
25/216
1/216
215/216
Find the expectation of a random variable X if the cdf .
-5
e-5
5
6
1/5
Compute the mean for continuous random variable X with probability density function .
2/3
0
1/3
1
Mean does not exist
If the variance of a random variable X is given Var(X)=3. Then Var(2X) is
12
6
3
1
9
Indicate the expectation of a Poisson random variable X with parameter .
0
Indicate the variance of a Poisson random variable X with parameter .
0
Indicate the formula for conditional expectation.
The table below shows the pmf of a random variable X. What is the Var(X)?
X |
-2 |
1 |
2 |
Р |
0,1 |
0,6 |
0,3 |
0.5
1.67
4.71
1.2
4.7
The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function . Calculate the probability that the lifetime of the machine part is less than 6.
0.04
0.15
0.47
0.53
0.94
The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function . Calculate the probability that the lifetime of the machine part is less than 5.
0.03
0.13
0.42
0.58
0.97
If Var(X)=2, find Var(-3X+4).
12
10
9
18
3
The table below shows the pmf of a random variable X. Find E[X] and Var(X).
X |
-1 |
0 |
1 |
P |
0.2 |
0.3 |
0.5 |
E[X]= 0,7; Var(X) =0.24
E[X]= 0,3; Var(X) =0.27
E[X]= 0,3; Var(X) =0.61
E[X]= 0,8; Var(X) =0.21
E[X]= 0,8; Var(X) =0.24
What kind of distribution is given by the density function ()?
Poisson distribution
Normal distribution
Uniform distribution
Bernoulli distribution
Exponential distribution