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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

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \textbf{Year}& \textbf{2026 - 1}& \textbf{2026 - 2}& \textbf{2025 - 1}& \textbf{2025 - 2}& \textbf{2024 - 1}& \textbf{2024 - 2}& \textbf{2023}& \textbf{2022}& \textbf{2021 - 1}& \textbf{2021 - 2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}\\\hline \textbf{1 Mark Count}&1&1&1&0&2&1&0&0&1&1&0&0.8&2\\\hline \textbf{2 Marks Count}&1&1&1&2&1&1&1&0&2&2&0&1.2&2\\\hline \textbf{Total Marks}&3&3&3&4&4&3&2&0&5&5&\bf{0}&\bf{3.2}&\bf{5}\\\hline \end{array}}}$$

Questions without answers in Probability

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Arrange the following intervals of standard normal variate in increasing order of probabilities.$0 \leq \mathrm{Z} \leq+1$$+1 \leq Z \leq+2$$+2 \leq Z \leq+3$$+3 \leq Z<+...
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Let ${N(t), t≥0}$ be a Poisson process with rate $λ=2$. Given that $N(3)=1$, the expected arrival time of the first event of the process is: a)1b)2/3​c)3/2​d)3
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Let $X_1, X_2, \ldots, X_n$ be independent random variables such that $$X_i \sim \mathcal{N}(0,1)$$Define the sample mean as\[\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i.\]...
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Let a fair coin is tossed $6$ times, where the coin tosses are independent of any other coin toss. Let $E_1$ be an event that there are at least $2$ heads in attempt $2,4...
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How to find the PMF of x1+x2 where x1,x2 are independent and identically distributed random variables with the uniform distribution on [0,1]
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In a school survey: 20% like all three activities (Music, Sports, Dance). 10% like none. No one likes only Dance. 60% of the students do not like Music. 70% of the studen...
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184
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Given below are two statements:$\text{Statement (I)}$ : Pearson's coefficient of correlation ' $r$ ' is a very useful measure of the strength and direction of the relatio...
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103
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If the input to an $\text{LTI}$ system is a random signal with a known mean and variance, the output will have:The same mean and variance as the input signalThe same mean...
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123
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A bag contains '$x$' red, '$x+5$' blue and '$x+7$' grey balls. If two balls are randomly drawn from the bag and the probability that both the balls are of same colour is ...
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146
146 views
The following data about the flow of liquid was observed in a continuous chemical process plantFlow rate(liters/sec)Frequency7.5 to 7.717.7 to 7.957.9 to 8.1358.1 to 8.31...
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209
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The map below shows all the routes connecting the locations $A, B, C$ and $D$. However, there are many bridges, each indicated with a (?) mark. Each bridge is open with p...
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170
170 views
Let x₁, x₂, ..., xₙ be a realization of a random sample of size n (≥ 2) from a N(μ, σ²) distribution, where -∞ < μ < ∞ and σ 0.Which of the following statements is/are t...
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170 views
A dice is thrown again and again until the sum of 5 is obtained. what is the expected number of throws?
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N5. A person \(A\) rolls a fair die until the outcome is 6. After that, \(B\) rolls the die as many times as \(A\) had rolled it. Compute the probability that \(B\) will ...
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203
203 views
Romeo and Juliet have a date at a given time, and each will arrive at the meeting place with a delay between 0 and 1 hour, with all pairs of delays being equally likely. ...
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369
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(b) A plane is missing, and it is presumed that it was equally likely to have gone down in any of three possible regions. Let 1-3, 1 denote the probability that the plane...
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176 views
Each round of a TV game show consists of ten questions. Before each round the host takes ten boxes and places prizes in nine of them, leaving one empty. The host then shu...
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159
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We select three points at random on the circumference of a circle. What is the probability that $\triangle A B C$ contains the centre $O$ in the interior?
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Let $A, B, C$ be events such that $P(A)=P(B)=P(C)=0.5, P(A \cap B)=0.3, P(A \cap C)=0$. Which of the following is/are true?$P(A \cup B)=0.75$$P(A \cup C)=1$$P(B \cap C)=0...
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278
278 views
Any file or resource that contains questions and examples to practice solving a problem about Shannon code theory. ANY HELP
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484
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$A$ and $B$ are playing a game which involves tossing a fair coin repeatedly. $A$ wins if the sequence $H H T$ appears before the sequence $H T H$. Compute the probabilit...
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479
479 views
The question is to find the distribution function of X.For the distribution to be valid the sum of all the probabilities must be equal to 1 , here its not equal to 1. So ...
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448
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A drawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark.(A) How many socks must he take out to be sure t...
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585
585 views
Consider the joint probability density function given by:$$f(x, y)=\begin{cases}    2xy & \text{if } 0 < x < 2 \text{ and } 0 < y < x \\    0 & \text{otherwise}\end{cases...
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414
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Can anyone please help me solve this question or explain the formula used in the solution part of the same?
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358
358 views
A company sometimes stops payments of quarterly dividends. If the company pays the quarterly dividend, the probability that the next one will be paid is $0.7$. If the com...
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261
261 views
Two defective bulbs are present in a set of five bulbs. To remove the two defective bulbs, the bulbs are chosen randomly one by one and tested. If $𝑋$ denotes the minimum...
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Let $𝑌_1 < 𝑌_2 < ⋯ < 𝑌_n$ be the order statistics of a random sample of size $𝑛$ from a continuous distribution, which is symmetric about its mean $𝜇$. Then the smallest ...
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