This set of Probability Multiple Choice Questions & Answers (MCQs) focuses on “Probability Distributions – Set 3”.

1. For which of the following conditions does the Geometric Random Variable P[X=n] = (1-p)^{n-1}p, hold?

a) Independent trials and n=0, 1, 2, 3, …

b) Independent trials and n=1, 2, 3, …

c) Dependent trials and n=0, 1, 2, 3, …

d) Dependent trials and n=1, 2, 3, …

View Answer

Explanation: The Geometric random variable always holds for independent trials and not dependent trials. It should also not start with 0. Hence the condition will hold for n=1, 2, 3 …

2. F(x) = (1/2)e^{-|x|}, -∞<x<∞, is a possible probability density function.

a) True

b) False

View Answer

Explanation: |x| is an even function;

|x|=x for all x≥0;

so, \(\int_{-∞}^{∞}f(x)dx=\int_{-∞}^{∞}e^{-|x|} dx = \int_{0}^{∞}e^{-x} dx=1,f(x)≥0,∀x\).

3. The set of points of discontinuity of a distribution function is __________

a) at least nonenumerable

b) at most nonenumerable

c) at least enumerable

d) at most enumerable

View Answer

Explanation: Any discontinuity in the distribution function can be mapped to a disjoint interval. The length of the interval is bound and hence countable. Thus the discontinuities are also countable i.e. enumerable.

4. The curve y=f(x) is called the distribution curve of the corresponding random variable x. It is evident that the distribution curve lies between ____________

a) Y=0, X=0

b) Y=0, X=1

c) Y=0, Y=1

d) X=0, X=1

View Answer

Explanation: Y is the distribution curve. Hence, it displays the probability of finding a certain point X. Therefore, Y can take any value between 0 and 1. Thus the curve lies between y=0 and y=1.

5. The number of changes of a stochastic process in a given interval of time follows _____________ law under certain conditions.

a) binomial

b) poisson

c) gamma

d) beta

View Answer

Explanation: The stochastic process can be represented as a counting process that displays a random number of points or events that have occurred until a particular time. The number of changes in the given interval is a Poisson variable dependent on time and another parameter.

6. The following is a probability mass function.

F(x)=\(\begin{Bmatrix}2 \,for \,x=\frac{1}{2}\\1 \,for \,x=\frac{1}{4}\\-1 \,for \,x=\frac{3}{4}\\0,elsewhere \end{Bmatrix}\)

a) True

b) False

View Answer

Explanation: f(\(\frac{3}{4}\)) = -1 < 0;

A probability function has to take values between 0 and 1 i.e. it can never be -1. Thus, f(x) is not a probability mass function.

7. For a two dimensional distribution function F(x, y), F(x, y) is ____________ for both the variable x and y.

a) monotonically decreasing

b) monotonically non-decreasing

c) monotonically increasing

d) monotonically non-increasing.

View Answer

Explanation: Let F(x, y), be denoted as a. Now, suppose a is positive. Then the 2-D function F(x, y), F(x, y) is also positive. Now, if a is negative, then a*a has to be positive. Thus again, F(x, y), F(x, y) is positive irrespective of the variables and the function.

8. The random variable is uniformly distributed in (0, 1). Then the probability distribution of Y = -2 logeX is ____________

a) Fy(y)=\(\frac{1}{2} e^{\frac{-1}{2} y}\), 0<y<∞

b) Fy(y)=\(\frac{1}{2} e^{\frac{1}{2} y}\), 0<y<∞

c) Fx(y)=\(\frac{1}{2} e^{\frac{-1}{2} y}\), 0<y<∞

d) Fy(y)=\(\frac{1}{2} e^{\frac{-1}{2}}\), 0<y<∞

View Answer

Explanation: Differentiating Y = -2 log

_{e}X, we get \(\frac{dy}{dx}=\frac{-2}{x}\). Now, \(\frac{dy}{dx}=\frac{-2}{x}\) <0 ∀x∈(0,1)

Hence, F

_{y}(y) = F

_{x}(x)\(|\frac{dy}{dx}|\) i.e. F

_{y}(y)=\(\frac{1}{2} e^{{-1}{2} y}\),0<y<∞

9. The salary of workers in a factory follows a binomial (300, 1/6) distribution. What will be the mean and standard deviation of the distribution curve?

a) 40, 41.6

b) 40, 6.4

c) 50, 41.6

d) 50, 6.4

View Answer

Explanation: For a binomial (n, p) distribution, the mean is (n*p) and the standard deviation is the square root of (n*p*(1-p)).

Now, n = 300, p = 1/6, 1-p = 5/6

Mean = (n*p) = 300 * 1/6 = 50

SD = square root of (50 * 5/6) = 6.4.

10. The graph of the standard normal distribution curve is?

a)

b)

c)

d)

View Answer

Explanation: For every standard normal distribution curve, the standard deviation has to be 1 and the mean or the average of points has to be zero. Thus, the curve must pass through the y-axis in the middle of the curve. Only curve (b) passes through the y-axis in the middle of the curve.

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