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Syllabus: Random variables, Uniform, Normal, Exponential, Poisson and Binomial distributions. Mean, median, mode and standard deviation. Conditional probability and Bayes theorem

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}& \textbf{2024-1} &\textbf{2024-2} &\textbf{2023} &\textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} &2&1&0& 0 &1&1&0&0.83&2
\\\hline\textbf{2 Marks Count} & 1&1&1&0 &2&2&0&1.16&2
\\\hline\textbf{Total Marks} & 4&3&2&0 &5&5&\bf{0}&\bf{3.16}&\bf{5}\\\hline
\end{array}}}$$

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#1
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 1
Given the probability mass function $p(X=x)$ ... $.$c=\frac{1}{3}$c=\frac{3}{2}$c=\frac{1}{2}$c=\frac{2}{3}$
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#2
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 2
Consider the random variables $(X, Y)$ ... 5,4.5)$.$\frac{1}{3}$\frac{2}{3}$\frac{5}{6}$\frac{1}{2}$
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#3
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 3
Consider a discrete random variable $X$ ... >0\right)$ ?$\frac{1}{4}$\frac{3}{8}$\frac{1}{2}$\frac{3}{4}$
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#4
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 4
Consider the following discrete random variable problem:A number $X$ is chosen at random from the set $\{1,2,3,4,5\}$. Subsequently, a second number $Y$ is chosen at random from the ... =5 \mid Y=5)=1$\mathbb{P}(X=5, Y=5):=\frac{1}{20}$
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#5
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 5
Suppose that $n$ people have their hats returned at random. Let $X_{i}$ be 1 if the $i$ ... n(n-1)}$\frac{2}{n(n-1)}$\frac{1}{n^{2}}$\frac{1}{n}$
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#6
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 6
The following is the probability distribution for a random variable, $X$,What is the variance of $X$ ?0.8000.4470.8941.225
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#7
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 7
James Bond is playing a game that involves repeatedly rolling a fair standard 6-sided die. What is the expected number of rolls until he gets a 5 ?
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#8
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 8
Imagine that you do not understand anything in this class. You would have to randomly guess the answer for the above 5 problems. What is the probability that you can get at least two ...
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#9
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 9
Let $X_{1}$ and $X_{2}$ be two Bernoulli random variables with the probability of success $p$. These variables are independent, where $X_{i}=0$ with probability $1-p$ ...
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#10
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 10
Let the joint p.m.f. of $X$ and $Y$ be defined by$f(i, j)=\frac{i+j}{32}$where $i=1,2$ and $j=1,2,3,4$. Find $P(X \leq 3-Y)$.$\frac{5}{32}$\frac{8}{32}$\frac{9}{32}$\frac{10}{32}$
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#11
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 11
Let $X$ be a random variable that takes values from 0 to 9 with equal probability $\frac{1}{10}$. Define the random variable $Y=X \bmod 3$.Calculate $P(Y=0)+P(Y=1)$.A. $\frac{3}{10}$B. $\frac{4}{10}$C $\frac{5}{10}$D. $\frac{7}{10}$
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#12
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 12
Suppose you are applying to graduate school and want to improve your GRE writing score. Your score on any given day can be either 4 , 5 , or 6 , each with equal probability. You plan to ... $\frac{4}{9}$\frac{5}{9}$\frac{2}{9}$
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#13
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 13
Suppose $X$ is the number of dust storms that occur on Mars next year. Assume that $X$ is a discrete uniform random variable that takes one of the 101 values in the range ... frac{1275}{101}$\frac{50}{101}$\frac{100}{101}$\frac{2550}{101}$
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#14
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 14
Suppose the discrete random variable $X$ has the probability mass function $f(x)=\frac{11-2 x}{24}, \quad x=1,2,3,4$.Find $E\left[X^{2}\right]$.$\frac{110}{24}$\frac{120}{24}$\frac{130}{24}$\frac{140}{24}$
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#15
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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 15
Let $X$ and $Y$ be independent discrete random variables with probability mass functions $P(X=$ $k)$ and $P(Y=k)$.What is $P(\min \{X, Y\} \leq x)$ ... x)$P(\min \{X, Y\} \leq x)=P(X>x)+P(Y>x)-P(X>x) P(Y>x)$
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#16
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IIT Madras MS DSAI Written Test 2024
Two fair 6-face diced are tossed independently. Let X be the random variable of the sum of two numbers on dices and let Y be the absolute difference of two numbers on dices. What is ... X $\geq$ 2Y)?a) 22/36b) 24/36c) 26/36c) 28/36d) 32/36
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#17
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David Stirzaker, Elementary Probability, Chapter 1, Example 1.9 Urn
An urn contains $n$ heliotrope and $n$ tangerine balls. A fair die with $n$ sides is rolled. If the $r^{th}$ face is shown, $r$ balls are removed ... a bag. What is the probability that a ball removed at random from the bag is tangerine?
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#18
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GATE DS&AI 2024 | Question: 1
Consider the following statements:The mean and variance of a Poisson random variable are equal.For a standard normal random variable, the mean is zero and the variance is one. ... $\text{(i)}$ and $\text{(ii)}$ are false
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#19
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GATE DS&AI 2024 | Question: 2
​​​​​Three fair coins are tossed independently. $T$ is the event that two or more tosses result in heads. $S$ is the event that two or more tosses result in tails.What is the probability of the event $T \cap S$?$0$0.5$0.25$1$
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#20
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GATE DS&AI 2024 | Question: 17
Let the minimum, maximum, mean and standard deviation values for the attribute income of data scientists be ₹$46000$, ₹ $170000$, ₹ $96000$, and ₹ $21000$, ... $0.476$0.623$2.304$
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