Normal Distribution properties

Q. If X is a normal variate with mean 30 and standard deviation 5, what is Probability $(26\le X\le 34),$. Given A (z = 0.8) = 0.2881, where A represents area.

1. 0.2881
2. 0.5762
3. 0.8181
4. 0.1616

Q.

Let $X_1,X_2\text{and}{X}_3$ be independent and identically distributed random variables with the uniform distribution on [0, 1]. The probability $P(X_1+X_2\le X_3)$ is the largest} is .

1. 0.16

Q.

What term refers to the standard deviation of the sampling distribution?

Q.

What is a normal probability curve?

Q. A simple random sample of 100 observations was taken from a large population. The sample mean & the standard deviation were determined to be 80 and 12 respectively. The standards error of mean is

1. 1.2

Q. Let X be a normal random variable with mean 1 and variance 4. The probability P(X < 0) is

1. 0.5
2. greater than zero and less than 0.5
3. greater than 0.5 and less than 1.0
4. 1.0

Q.

Let X be a zero mean unit variance Gaussian random variable. E[|X|] is equal to .

1. 0.7978

Q. The annual precipitation data of a city is normally distributed with mean and standard deviation a 1000 mm and 200 mm, respectively. The probability that the annual precipitation will be more than 1200 mm is

1. <50%
2. 50%
3. 75%
4. 100%

Q. In a sample of 100 students, the mean of the marks (only integers) obtained by them in a test is 14 with its standard deviation of 2.5 (marks obtained can be fitted with a normal distribution). The percentage of students scoring 16 marks is

(Area under standard normal curve between z = 0 and z = 0.6 is 0.2257; and between z = 0 and z = 1.0 is 0.3413)

1. 36
2. 23
3. 12
4. 10

Q. Suppose X is a normal random variable with mean 0 and variance 4. Then the mean of the absolute value of X is

1. $\frac{1}{\sqrt{2\pi }}$
2. $\frac{2\sqrt{2}}{\sqrt{\pi }}$
3. $\frac{2\sqrt{2}}{\pi }$
4. $\frac{2}{\sqrt{\pi }}$

Q. For a random variable $x(-\infty <x<\infty )$ following normal distribution, the mean is $\mu =100.$ If the probability is $P=\alpha \text{for}x\ge 110.$ Then the probability of x lying between 90 and 110 i.e. $P(90\le x\le 110)$ and equal to

1. $1-2\alpha$
2. $1-\alpha$
3. $1-\alpha /2$
4. $2\alpha$

Q. The standard normal probability function can be approximated as
$F\left(X_N\right)=\frac{1}{1+exp(-1.7255X_N|X_N{|}^{0.12})},\text{where}{X}_N=$ standard normal deviate. If mean and standard deviation of annual precipitation are 102 cm and 27 cm respectively, the probability that the annual precipitation will be between 90 cm and 102 cm is

1. 66.7%
2. 50.0%
3. 33.3%
4. 16.7%

Q.

A nationalized bank has found that the daily balance available in its savings accounts follows a normal distribution with a mean of Rs. 500 and a standard deviation of Rs. 50. The percentage of savings account holders, who maintain an average daily balance more than Rs. 500 is ______.

1. 50

Q.

Which one of the given measures of dispersion is considered best?

1. Standard deviation

2. Range

3. Mean deviation

4. Coefficient of variation

Q. If the standard deviation of the spot speed of vehicles in a highway is 8.8 km/h and the mean speed of the vehicles is 33 km/h, the coefficient of variation in speed is

1. 0.1517
2. 0.1867
3. 0.2666
4. 0.3646

Q. Consider a Gaussian distributed random variable with zero mean and standard deviation $\sigma .$ The value of its cumulative distribution function at the origin will be

1. 0
2. 0.5
3. 1
4. $10\sigma$

Q. Let U and V be two independent zero mean Gaussian random variables of variances $\frac{1}{4}\text{and}\frac{1}{9}$ respectively. The probability $P(3V\ge 2U)\text{is}$

1. $\frac{4}{9}$
2. $\frac{1}{2}$
3. $\frac{2}{3}$
4. $\frac{5}{9}$

Q.

The area (in percentage) under standard normal distribution curve of random variable Z within limits from -3 to +3 is _____.

1. 99.7

Q.

Let X be a Gaussian random variable mean 0 and variance ${\sigma }^2.$ Let Y = max(X, 0) where max (a, b) is the maximum of a and b. Th emedian of Y is

1. 0

Q. $\text{Let}{X}_1,X_2$ be two independent normal random variables with means ${\mu }_1,{\mu }_2$ and standard deviations ${\sigma }_1,{\sigma }_2,\text{respectively. Consider}Y=X_1-X_2,{\mu }_1={\mu }_2=1,{\sigma }_1=1,{\sigma }_2=2.\text{Then}$

1. Y is normally distributed with mean 0 and variance 1
2. Y is normally distributed with mean 0 and variance 5
3. Y has mean 0 and variance 5, but is NOT normally distributed
4. Y has mean 0 and variance 1, but is NOT normally distributed

Q. If f(x) is a continous real valued random variable defined over the interval $(-\infty ,+\infty )$ and its occurance is defined by the density function given as $f\left(x\right)=\frac{1}{\sqrt{2\pi }b}exp\{-\frac{1}{2}{\left(\frac{x-a}{b}\right)}^2\}$ where a and b are the statistical attributes of the random variable {x}. The value of the integral ${\int }_{-\infty }^a\frac{1}{\sqrt{2\pi }b}exp\{-\frac{1}{2}{\left(\frac{x-a}{b}\right)}^2\}dx\text{is}$

1. 1
2. 0.5
3. $\pi$
4. $\pi /2$

Q.

For a construction project the mean and standard deviation of the completion time are 200 days and 6.1 days respectively. Assume normal distribution and use the value of standard normal deviate z = 1.64 for the 95% confidence level. The maximum time required (in days) for the completion of the project would be

1. 210

Q.

What is a standard normal probability distribution?

Q. A normal random variable X has the following probability density function

$f_x\left(x\right)=\frac{1}{\sqrt{8\pi }}{e}^{-\left\{\frac{(x-1{)}^2}{8}\right\}},-\infty <x<\infty$

Then ${\int }_1^{\infty }{f}_x\left(x\right)dx?$

1. 0
2. $\frac{1}{2}$
3. $1-\frac{1}{e}$
4. 1

Q.

What is the role of sample size in the central limit theorem?

Q. The mean square of a zero mean random process is kT/C, where k is Boltzman's constant. T is the absolute temperature, and C is capacitance. The standard deviation of the random.

1. $\frac{C}{kT}$
2. $\sqrt{\frac{kT}{C}}$
3. $\frac{kT}{C}$
4. $\frac{\sqrt{kT}}{C}$

Q.

What is the difference between standard error and standard deviation?

Q. Number of tigers in a reserve is normally distributed with mean and variance respectively as $1200and9\times {10}^4$. The probability of finding more than 1800 tigers is approximately.

1. 0.0125
2. 0.025
3. 0.05
4. None of these

Q. An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

1. 0.067
2. 0.073
3. 0.082
4. 0.091

Q.

A function f(x) is designed as to be a valid probability density function the value of A must be

$f\left(x\right)=\{\begin{array}{cc}0& ;x<3\\ \frac{1}{A}(3x+5)& ;3<x<6.\text{For}f\left(x\right)\\ 0& ;x>6\end{array}$

1. 55.5
2. 40.5
3. 50
4. 63.5