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**Q. 1 The probability density function of a continuous random variable distributed uniformly between x and y (for y > x) is [2019 : 2 Marks, Set-II]**

**(a) y - x(b) **

**(c) ) x - y**

**(d) **

**Ans: (b)**

Probability density function of uniform distribution is

**Q. 2 Probability (up to one decimal place) of consecutively picking 3 red balls without replacement from a box containing 5 red balls and 1 white ball is _______ . [2018 : 1 Mark, Set-II]****Ans (0.5)**

**Probability **

**Q. 3 The graph of a funotion f{x) is shown in the figure**

**For f(x) to be valid probability density function, the value of h is **

** [2018 : 1 Mark, Set-II]****(a) 1/3****(b) 2/3****(c) ****1 ****(d) 3**

**Ans (a)**

**Q. 4 A probability distribution with right skew is shown in the figure.**

**The correct statement for the probability distribution is ****(a) Mean is equal to mode****(b) Mean is greater than median but less than mode ****(c) Mean is greater than median and mode****(d) Mode is greater than median. [2018 :1 Mark, Set-II]**

**Ans **(c)

t_{L} < f_{mea }= Curve is skew to right,

mode < mean

i.e., mean > median and mode

Mean is greater than the mode and the median. This is common for a distribution that is skewed, to the right [i.e., bunched up toward the left and a ‘tail’ stretching toward the right].

**Q. 5 For the function f(x) = a + bx, 0 ≤ x ≤ 1, to be a valid probability density function, which one of the following statements is correct?****(a) a = 1, b = 4****(b) a = 0.5, b = 1****(c) a = 0, b = 1****(d)a = 1, b = 1 [2017 : 2 Marks, Set-I]**

**Ans: **(b)

**Q. 6 The number of parameters in the univariate exponential and Gaussian distributions, respectively, are **

**(a) 2 and 2****(b) 1 and 2****(c) 2 and 1****(d) 1 and 1 [2017 : 1 Mark, Set-I]****Ans (b)**

In exponential,

x = 0

The parameter is λ.

In Gaussin

The parameters are µ and α.

Therefore, answer is (b).

**Q. 7 If f(x) arid y(x) are two probability density functions,**

**Which one of the following statements is true?****(a) Mean of f(x) and g(x) are same; Variance of f{x) and g(x) are same**

**(b) Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are different**

**(c) Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are same**

**(d) Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are different**

**Ans (b)**

Mean of f(x) is t(x)

Variance of f(x) is E(x)^{2} - {E(x)^{2}} where

⇒ Variance is a^{3}/6

Next, mean of g(x) is E(x)

Variance of g(x) is E(x^{2}) - {E(x)}^{2}, where,

⇒ Variance is a^{3}/2

∴ Mean of f(x) and g(x) are same but variance of f(x) and g(x) are different

**Q. 8 X and T are two random independent events. It is known that P (X ) = 0.40 and P { X ∪ Y ^{c} ) = 0.7. Which one of the following is the value of P (X ∪ Y)? [2016 : 1 Mark, Set-II](a) 0.7(b) 0.5(c) 0.4(d) 0.3**

**Ans **(a)

(Since X, Y are independent events)

**Q. 9 Probability density function of a random variable X is given below**

** [2016 : 2 Marks, Set-I]**

**(a) 3/4**

**(b) 1/4**

**(c) 1/4**

**(d) 1/8****Ans **(a)

**Q. 10 The probability density function of a random variable, x is**

**Ans:**

**The mean, μ _{x} of the random variable is________ [2015 : 2 Marks, Set-11]**

**Ans: **1.066

**Q. 11 Consider the following probability mass function (p.m.f.) of a random variable X.**

**if q = 0.4, the variance of X is __________**

**Ans: **0.24

Given

Required value = V(X) = E{X^{2}) - [E{X)]^{2}

**Q. 12 A traffic office im poses on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is __________ . [2014 : 2 Marks, Set-I**

**Ans: 0.265**

Mean λ, = 5

P(x < 4) = { p(x = 0) + p(x = 1) + p(x = 2) + p(x = 3)}

**Q. 13 A fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes: (i) Head, (ii) Head, (iii) Head, (iv) Head. The probability of obtaining a 'Tail' when the coin is tossed again is**

**(a) 0****(b) 1/2****(c) 4/5****(d) 1/5****Ans: b**

**Q. 14 The probability density function of evaporation E on any day during a year in a watershed is given by **

**The probability that Elies in between 2 and 4 mm/day in a day in the watershed is (in decimal)_____ . [2014 : 1 Mark, Set-I]**

**Ans: **0.4

**Q. 15 Find the value of λ such that function f(x) is valid probability density function f(x) = λ(x - 1) ( 2 - x) for 1 ≤ x ≤ 2 = 0 otherwise [2013 : 2 Marks]**

**Ans: 6**

**Q. 17 In an experiment, positive and negative values are equally likely to occur. The probability of obtaining at most one negative value in five trials is (a) 1/32 **

**(b) 2/32 **

**(c) 3/32 **

**(d) 6/32 [2011 : 2 Marks**

**Ans: d**

Since negative and positive are equally likely, the distribution of number of negative values is binomial with n = 5 and p = 1/2

Let X represent number of negative values in 5 trials.

p(at most 1 negative value

**Q. 18 The annual precipitation data of a city is normally distributed with mean and standard deviation as 1000 mm and 200 mm, respectively. The probability that the annual precipitation will be more than 1200 mm is****(a) <50%****(b) 50%****(c) 75%****(d) 100% [2011 : 2 Marks]**

**Ans: **a

The annual precipitation is normally distributed with μ = 1 000 mm and σ = 200 mm

Where z is the standard normal variate.

In normal distribution Now, since p(-1 < z < 1) ≈ 0.68

(≈ 68% of data is within one standard deviation of mean)

**Q. 19 There are two containers, with one containing 4 red and 3 green balls and the other containing 3 blue and 4 green balls. One ball is drawn at random from each container. The probability that one of the balls is red and the other is blue will be****(a) 1/7****(b) 9/49****(c) 12/49****(d) 3/7 [2011 : 1 Mark]****Ans: c**

p(one ball is Red & another is blue)

= p(first is Red and second is Blue)

**Q. 20 Two coins are simultaneously tossed. The probability of two heads sim ultaneously appearing is [2010 : 1 Mark](a) 1/8(b) 1/6 (c) 1/4(d) 1/2Ans: c**

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