Question 1 |

The shape of the cumulative distribution function of Gaussian distribution is

Horizontal line | |

Straight line at 45 degree angle | |

Bell-shaped | |

S-shaped |

Question 1 Explanation:

PDF:f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-(x-\mu )^2/(2\sigma ^2)}

CDF:F(x)=\frac{1}{2}\left [ 1+eff\left ( \frac{x-\mu }{\sigma \sqrt{2}} \right ) \right ]

Question 2 |

A fair (unbiased) coin is tossed 15 times. The probability of getting exactly 8 Heads
(round off to three decimal places), is ________.

0.523 | |

0.421 | |

0.196 | |

0.223 |

Question 2 Explanation:

P(H)=\frac{1}{2}

P(T)=\frac{1}{2}

Probability of getting exactly 8 heads out of 15 trial ={^{15}C_8}\left [ \frac{1}{2} \right ]^8 \times \left [ \frac{1}{2} \right ]^{15-8}=0.196

P(T)=\frac{1}{2}

Probability of getting exactly 8 heads out of 15 trial ={^{15}C_8}\left [ \frac{1}{2} \right ]^8 \times \left [ \frac{1}{2} \right ]^{15-8}=0.196

Question 3 |

The probability density function of a continuous random variable distributed uniformly between x and y (for y \gt x) is

\frac{1}{x-y} | |

\frac{1}{y-x} | |

x-y | |

y-x |

Question 3 Explanation:

Probability density function of uniform distribution is f(x)=\frac{1}{y-x}

Question 4 |

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 ______

0 | |

0.25 | |

0.5 | |

1 |

Question 4 Explanation:

Probability, \bar{P}=\frac{5}{6}\times \frac{4}{5}\times \frac{3}{4}=\frac{1}{2}=0.5

Question 5 |

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

The correct statement for the probability distribution is

The correct statement for the probability distribution is

Mean is equal to mode | |

Mean is greater than median but less than mode | |

Mean is greater than median and mode | |

Mode is greater than median |

Question 5 Explanation:

t_{L}\lt t_{mean}=Curve is skew to right.

Mode \lt mean

i.e., Mean \gt 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].

Question 6 |

The graph of a function f(x) is shown in the figure.

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

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

0.33 | |

0.66 | |

1 | |

3 |

Question 6 Explanation:

\begin{aligned} \int_{0}^{3}f\left ( x \right )dx &=1 \\ \int_{0}^{1}f\left ( x \right )dx+\int_{1}^{2}f\left ( x \right )dx+\int_{2}^{3}f\left ( x \right )dx &=1 \\ \frac{h}{2}+\frac{2h}{2+\frac{3h}{2}} &=1 \\ 6h &=2 \\ \Rightarrow\; \; h &=\frac{1}{3} \end{aligned}

Question 7 |

A two-faced fair coin has its faces designated as head (H) and tail (T). This coin is tossed three times in succession to record the following outcomes: H, H, H. If the coin is tossed one more time, the probability (up to one decimal place) of obtaining H again, given the previous realizations of H, H and H, would be______

0.25 | |

0.5 | |

0.75 | |

0.2 |

Question 8 |

If f(x) and g(x) are two probability density functions,

f(x)=\left\{\begin{matrix} \frac{x}{a}+1 & :-a\leq x \lt 0\\ -\frac{x}{a}+1 & :0\leq x\leq a\\ 0& : otherwise \end{matrix}\right.

g(x)=\left\{\begin{matrix} -\frac{x}{a} & :-a\leq x \lt 0\\ \frac{x}{a} & :0\leq x\leq a\\ 0& : otherwise \end{matrix}\right.

Which one of the following statements is true?

f(x)=\left\{\begin{matrix} \frac{x}{a}+1 & :-a\leq x \lt 0\\ -\frac{x}{a}+1 & :0\leq x\leq a\\ 0& : otherwise \end{matrix}\right.

g(x)=\left\{\begin{matrix} -\frac{x}{a} & :-a\leq x \lt 0\\ \frac{x}{a} & :0\leq x\leq a\\ 0& : otherwise \end{matrix}\right.

Which one of the following statements is true?

Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are same. | |

Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are different. | |

Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are same. | |

Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are different. |

Question 8 Explanation:

Mean of f(x) is E(x)

=\int_{-a}^{0}x\left ( \frac{x}{a}+1 \right )dx+\int_{0}^{a}x\left ( \frac{-x}{a}+1 \right )dx

=\left ( \frac{x^{3}}{3a}+\frac{x^{2}}{2} \right )_{-a}^{0}+\left ( \frac{-x^{3}}{3a}+\frac{x^{2}}{2} \right )_{0}^{a}=0

Variance of f(x) is E\left ( x \right )^{2}-\left \{ E\left ( x \right )^{2} \right \} where,

E\left ( x \right )^{2}=\int_{-a}^{0}x^{2}\left ( \frac{x}{a}+1 \right )dx+\int_{0}^{a}x^{2}\left ( \frac{-x}{a}+1 \right )dx

=\left ( \frac{x^{4}}{4a}+\frac{x^{3}}{3} \right )_{-a}^{0}+\left ( \frac{-x^{4}}{4a}+\frac{x^{3}}{3} \right )_{0}^{a}=\frac{a^{3}}{6}

\Rightarrow \; \; Variance is \frac{a^{3}}{6}

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

=\int_{a}^{0}s\left ( \frac{-x}{a} \right )dx+\int_{0}^{a}x\left ( \frac{x}{a} \right )dx=0

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

E\left ( x^{2} \right )=\int_{-a}^{0}x^{2}\left ( \frac{-x}{a} \right )dx+\int_{0}^{a}x^{2}\left ( \frac{x}{a} \right )dx=\frac{a^{3}}{2}

\Rightarrow \; \; Variance is \frac{a^{3}}{2}

\therefore Mean of f(x) and g(x) are same but variance of f(x) and g(x) are different.

=\int_{-a}^{0}x\left ( \frac{x}{a}+1 \right )dx+\int_{0}^{a}x\left ( \frac{-x}{a}+1 \right )dx

=\left ( \frac{x^{3}}{3a}+\frac{x^{2}}{2} \right )_{-a}^{0}+\left ( \frac{-x^{3}}{3a}+\frac{x^{2}}{2} \right )_{0}^{a}=0

Variance of f(x) is E\left ( x \right )^{2}-\left \{ E\left ( x \right )^{2} \right \} where,

E\left ( x \right )^{2}=\int_{-a}^{0}x^{2}\left ( \frac{x}{a}+1 \right )dx+\int_{0}^{a}x^{2}\left ( \frac{-x}{a}+1 \right )dx

=\left ( \frac{x^{4}}{4a}+\frac{x^{3}}{3} \right )_{-a}^{0}+\left ( \frac{-x^{4}}{4a}+\frac{x^{3}}{3} \right )_{0}^{a}=\frac{a^{3}}{6}

\Rightarrow \; \; Variance is \frac{a^{3}}{6}

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

=\int_{a}^{0}s\left ( \frac{-x}{a} \right )dx+\int_{0}^{a}x\left ( \frac{x}{a} \right )dx=0

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

E\left ( x^{2} \right )=\int_{-a}^{0}x^{2}\left ( \frac{-x}{a} \right )dx+\int_{0}^{a}x^{2}\left ( \frac{x}{a} \right )dx=\frac{a^{3}}{2}

\Rightarrow \; \; Variance is \frac{a^{3}}{2}

\therefore Mean of f(x) and g(x) are same but variance of f(x) and g(x) are different.

Question 9 |

X and Y are two random independent events. It is known that P(X)=0.40 and
P(X\cup Y^{C})=0.7. Which one of the following is the value of P(X\cup Y) ?

0.7 | |

0.5 | |

0.4 | |

0.3 |

Question 9 Explanation:

\; \; \; \; P\left ( X\: \cup \: Y^{c} \right )=0.7

\Rightarrow \; \; P\left ( X \right )+P\left ( Y^{c} \right )-P\left ( X \right )P\left ( Y^{c} \right )=0.7

(Since X, Y are independent events)

\Rightarrow \; \; P\left ( X \right )+1-P\left ( Y \right )-P\left ( X \right )\left \{ 1-P\left ( Y \right ) \right \}=0

\Rightarrow \; \; P\left ( X \right )-P\left ( X\: \cap \: Y \right )=0.3\; \; \; \; \; \; ...\left ( i \right )

\; \; \; \; P\left ( X\: \cup \: Y \right )=P\left ( X \right )+P\left ( Y \right )-P\left ( X\: \cap \: Y \right )

\; \; \; \; =0.4+0.3=0.7

\Rightarrow \; \; P\left ( X \right )+P\left ( Y^{c} \right )-P\left ( X \right )P\left ( Y^{c} \right )=0.7

(Since X, Y are independent events)

\Rightarrow \; \; P\left ( X \right )+1-P\left ( Y \right )-P\left ( X \right )\left \{ 1-P\left ( Y \right ) \right \}=0

\Rightarrow \; \; P\left ( X \right )-P\left ( X\: \cap \: Y \right )=0.3\; \; \; \; \; \; ...\left ( i \right )

\; \; \; \; P\left ( X\: \cup \: Y \right )=P\left ( X \right )+P\left ( Y \right )-P\left ( X\: \cap \: Y \right )

\; \; \; \; =0.4+0.3=0.7

Question 10 |

The spot speeds (expressed in km/hr) observed at a road section are 66, 62, 45, 79, 32, 51, 56, 60, 53, and 49. The median speed (expressed in km/hr) is ________.

(Note: answer with one decimal accuracy)

(Note: answer with one decimal accuracy)

54.5 | |

51.5 | |

53.5 | |

56 |

Question 10 Explanation:

Median speed is the speed at the middle value in series of spot speeds that are arranged in ascending order. 50% of speed values will be greater than the median 50% will be less than the median.

Ascending order order of spot speed studies are 32, 39, 45, 51, 53, 56, 60, 62, 66, 79

Median speed=\frac{53+56}{2}=54.5 km/hr

Ascending order order of spot speed studies are 32, 39, 45, 51, 53, 56, 60, 62, 66, 79

Median speed=\frac{53+56}{2}=54.5 km/hr

There are 10 questions to complete.