1.
Expected value is also called as __________
Correct Answer
A. Mean
Explanation
Expected value is also called as mean. The expected value represents the average outcome of a random variable over a large number of trials. It is calculated by multiplying each possible outcome by its probability and summing them up. The mean is a measure of central tendency and provides information about the average value or expected outcome of a given variable. Therefore, the correct answer for this question is mean.
2.
Standard deviation is _______ of variance.
Correct Answer
A. Square root
Explanation
The standard deviation is the square root of the variance. Variance measures the average of the squared differences from the mean, while the standard deviation is the square root of the variance and represents the average amount of deviation from the mean. Therefore, the standard deviation is the square root of the variance.
3.
If p(x) is probability mass function and p(x) = k, 2k, 3k, 4k, 5k then k = __________
Correct Answer
A. 1/15
Explanation
The given probability mass function states that the probabilities of different outcomes are in the form of k, 2k, 3k, 4k, and 5k. In a probability mass function, the sum of all probabilities must equal 1. Therefore, we can set up the equation k + 2k + 3k + 4k + 5k = 1. Simplifying the equation, we get 15k = 1. Dividing both sides by 15, we find that k = 1/15.
4.
If x = 0,1,2,3,4 and p(x) = 0.1, 0.2, 0.3, 0.15, 0.25. Then p(x<1) = _____
Correct Answer
A. 0.1
Explanation
The probability of x being less than 1 can be calculated by summing up the probabilities of x being 0 and x being 1. From the given information, p(0) = 0.1 and p(1) = 0.2. Therefore, p(x
5.
In discrete uniform distribution E(X) = ________
Correct Answer
A. (n+1)/2
Explanation
In a discrete uniform distribution, all outcomes have an equal probability. The expected value, E(X), represents the average value of the random variable X. In this case, the random variable X can take on values from 1 to n with equal probability. The formula for the expected value of a discrete uniform distribution is (n+1)/2, where n represents the number of possible outcomes. Therefore, in this case, the correct answer is (n+1)/2.
6.
If x = 0,1,2,3,4 and p(x) = 0.1, 0.2, 0.3, 0.15, 0.25. Then p(x>=3) = _____
Correct Answer
A. 0.4
Explanation
The probability function p(x) represents the probability of a certain value of x occurring. In this case, x can take on the values 0, 1, 2, 3, or 4 with corresponding probabilities of 0.1, 0.2, 0.3, 0.15, and 0.25 respectively. The question asks for the probability of x being greater than or equal to 3, which includes the values 3 and 4. The probability of x being 3 is 0.15 and the probability of x being 4 is 0.25. Adding these probabilities together gives 0.4, which is the probability of x being greater than or equal to 3. Therefore, the correct answer is 0.4.
7.
If p(x) is probability mass function where p(x) = c, 2c, 4c, 2c, c then c = ______
Correct Answer
A. 1/10
Explanation
The given probability mass function assigns probabilities to the values x as c, 2c, 4c, 2c, and c. Since the sum of all probabilities in a probability mass function must equal 1, we can set up the equation c + 2c + 4c + 2c + c = 1. Simplifying this equation gives us 10c = 1. Dividing both sides by 10 gives us c = 1/10. Therefore, the value of c is 1/10.
8.
In Binomial distribution p is called probability of _______
Correct Answer
A. Success
Explanation
In Binomial distribution, p represents the probability of success. This means that p is the likelihood of achieving a desired outcome or event in a series of independent trials. The binomial distribution is used to model situations where there are only two possible outcomes, success or failure, in each trial. Therefore, p is the probability associated with the occurrence of the desired outcome, which is success.
9.
In discrete uniform distribution var(X) = ________
Correct Answer
A. ( n^{2} - 1 ) /12
Explanation
In discrete uniform distribution, var(X) represents the variance of the random variable X. The formula for variance in a discrete uniform distribution is (n^2 - 1) / 12, where n is the number of possible outcomes. This formula is derived from the properties of a discrete uniform distribution, where all outcomes have equal probabilities. The variance measures the spread or variability of the distribution, and in this case, it is calculated by taking the difference between each outcome and the mean, squaring those differences, and then averaging them. Hence, the correct answer is (n^2 - 1) / 12.
10.
In Binomial distribution q can be find using formula _____
Correct Answer
A. 1-p
Explanation
The formula to find q in the binomial distribution is 1-p. This is because q represents the probability of the event not occurring, while p represents the probability of the event occurring. Since the sum of the probabilities of an event occurring and not occurring is always 1, we can find q by subtracting p from 1.
11.
In Binomial distribution E(X) = __________
Correct Answer
A. N*p
Explanation
The expected value of a binomial distribution is given by the formula n*p, where n represents the number of trials and p represents the probability of success in each trial. This formula calculates the average value or mean of the distribution.
12.
In Binomial distribution var(X) = __________
Correct Answer
A. N*p*q
Explanation
The variance of a binomial distribution is equal to the product of the number of trials (n), the probability of success (p), and the probability of failure (q). This formula is derived from the properties of the binomial distribution, where the variance measures the spread or variability of the distribution.
13.
In Poisson distribution E(X) = _______
Correct Answer
A. M
Explanation
The expected value of a Poisson distribution is denoted by E(X) and is equal to the mean of the distribution, which is represented by the variable m. Therefore, the correct answer is m.
14.
In Poisson distribution mean = _______
Correct Answer
A. Variance
Explanation
In Poisson distribution, the mean is equal to the variance. This means that the average number of events occurring in a given time period is also equal to the measure of the spread or variability of those events. Therefore, the correct answer is "variance".
15.
In Exponential distribution E(X) = _______
Correct Answer
A. 1/λ
Explanation
The exponential distribution represents the time between consecutive events in a Poisson process, where events occur at a constant average rate λ. The expected value or mean of the exponential distribution is denoted by E(X) and is equal to 1/λ. This means that on average, we can expect an event to occur every 1/λ units of time.
16.
In Exponential distribution Var(X) = ______
Correct Answer
A. 1/(λ)^{2}
Explanation
The correct answer is 1/(λ)2. In the exponential distribution, the variance of a random variable X is equal to the reciprocal of the square of the rate parameter (λ). Therefore, the variance is given by 1/(λ)2.
17.
In Continuous uniform distribution E(X) = _________
Correct Answer
A. (b+a)/2
Explanation
The formula for the expected value (E(X)) in a continuous uniform distribution is (b+a)/2, where a and b are the lower and upper limits of the distribution. This formula calculates the average value of the random variable X over the entire range of possible values. In a continuous uniform distribution, all values within the range have an equal probability of occurring, so the expected value is simply the average of the lower and upper limits.
18.
In Continuous uniform distribution Var(X) = ________
Correct Answer
A. (b-a)^{2}/12
Explanation
In continuous uniform distribution, the variance of a random variable X is given by (b-a)2/12, where a represents the lower limit of the distribution and b represents the upper limit. This formula is derived from the properties of the continuous uniform distribution and represents the spread or variability of the data within the given range.
19.
The normal curve in Normal distribution is symmetrical about ______
Correct Answer
A. Mean
Explanation
The normal curve in a normal distribution is symmetrical about the mean. This means that the curve is equally distributed on both sides of the mean, creating a bell-shaped curve. The mean represents the average value of the data set, and in a normal distribution, it acts as the central point around which the data is distributed. Therefore, the normal curve is symmetrical about the mean.
20.
In Bisection method first approximation is ________
Correct Answer
A. (a+b)/2
Explanation
The first approximation in the Bisection method is calculated by taking the average of the initial interval endpoints, which is represented by (a+b)/2.
21.
Forward difference operator is denoted as ______
Correct Answer
A. Δ
Explanation
The forward difference operator is denoted as Δ. This symbol represents the change or difference between consecutive values in a sequence or function. It is commonly used in calculus and discrete mathematics to approximate derivatives and analyze rates of change. The Δ symbol is derived from the Greek letter delta, which is often used to represent a difference or change in mathematical notation.
22.
The ______ of the Normal distribution is at the middle of the curve which divides area into the equal parts.
Correct Answer
A. Mean
Explanation
The mean of the Normal distribution is the average value and represents the center of the curve. It is the point that divides the area under the curve into equal parts on both sides. The mean is a measure of central tendency and provides information about the average value of the data. In the Normal distribution, the mean is also the location of the peak of the curve.
23.
If p(x) = 0.5, -0.1, 0.6, 0. Check that it is a probability mass function or not.
Correct Answer
A. Not a probability mass function
Explanation
A probability mass function (PMF) assigns probabilities to discrete random variables. In this case, the values given (-0.1 and 0.6) are not valid probabilities because they are outside the range of 0 to 1. Therefore, the given values do not form a valid PMF and the correct answer is "Not a probability mass function".
24.
A random variable is said to be __________ if it takes any value in a given interval.
Correct Answer
A. Continuous
Explanation
A random variable is said to be continuous if it can take any value within a given interval. This means that there are infinite possible values that the random variable can assume, and it can take on any value within the specified range. In contrast, a discrete random variable can only take on specific, separate values. The terms "exponential" and "uniform" are not relevant to this question and do not describe the property of a random variable taking any value within an interval.
25.
A continuous random variable x is said to follow _____________, if it satisfies the probability density function f(x) = λ e ^{-λx}.
Correct Answer
A. Exponential
Explanation
A continuous random variable x is said to follow the exponential distribution if it satisfies the probability density function f(x) = λe^(-λx). The exponential distribution is commonly used to model the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. The parameter λ represents the rate at which events occur, and the exponential distribution is often used in various fields such as reliability analysis, queuing theory, and survival analysis.
26.
In Binomial distribution q is called probability of ________.
Correct Answer
A. Failure
Explanation
In binomial distribution, q represents the probability of failure. This means that q is the likelihood of an event not occurring or being unsuccessful. In the context of binomial distribution, q is used to calculate the probability of a certain number of failures or unsuccessful outcomes in a given number of trials or experiments. Therefore, the correct answer is "failure".
27.
If x = 0,1,2,3,4 and p(x) = 0.1, 0.2, 0.3, 0.15, 0.25. Then p(x=4) = _____
Correct Answer
A. 0.25
Explanation
The probability function p(x) gives the probability of each value of x. In this case, when x=4, p(x) is equal to 0.25. Therefore, the probability of x being equal to 4 is 0.25.
28.
If x = 0,1,2,3,4 and p(x) = 0.1, 0.2, 0.3, 0.15, 0.25. Then p(x>=3 & x<4) = _____
Correct Answer
A. 0.15
Explanation
The probability function p(x) represents the probability of x taking on a certain value. In this case, p(x) is given for x = 0,1,2,3,4. To find p(x>=3 & x
29.
If x = 0,1,2,3,4 and p(x) = 0.1, 0.2, 0.3, 0.15, 0.25. Then p(x>=1 & x<4) = _____
Correct Answer
A. 0.65
Explanation
The probability of x being greater than or equal to 1 and less than 4 can be calculated by summing the probabilities of x being 1, 2, and 3. Therefore, p(x>=1 & x