# Recent questions tagged random-variable 4 votes
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For a given biased coin, the probability that the outcome of a toss is a head is $0.4$. This coin is tossed $1,000$ times. Let $X$ denote the random variable whose value is the number of times that head appeared in these $1,000$ tosses. The standard deviation of $X$ (rounded to $2$ decimal place) is _________
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The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter $2$. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to $2$ decimal places) is ____________.
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Consider the two statements. $S_1:\quad$ There exist random variables $X$ and $Y$ such that $\left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2>\textsf{Var}[X]\textsf{Var}[Y]$ $S_2:\quad$ For all random variables $X$ ... Both $S_1$ and $S_2$ are true $S_1$ is true, but $S_2$ is false $S_1$ is false, but $S_2$ is true Both $S_1$ and $S_2$ are false
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$\text{Description for the following question:}$ If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then $\mathbb{P}(Z\leq a)=\int^a _{-\infty} \frac{\exp\{\frac{-z^2}{2}\}}{\sqrt{2\pi}}dz=\Phi(a),$ where $\Phi(a=-2)=0.02,$ ... less than 0.02 $\int^{18} _{-\infty} \frac{1}{\sqrt{2\pi .2^2}}\exp\{-\frac{1}{2}(\frac{x-24}{2})^2\}dx$
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$\text{Description for the following question:}$ If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then $\mathbb{P}(Z\leq a)=\int^a _{-\infty} \frac{\exp\{\frac{-z^2}{2}\}}{\sqrt{2\pi}}dz=\Phi(a),$ where $\Phi(a=-2)=0.02,$ ... more than 0.4 less than 0.5
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$\text{Description for the following question:}$ If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then $\mathbb{P}(Z\leq a)=\int^a _{-\infty} \frac{\exp\{\frac{-z^2}{2}\}}{\sqrt{2\pi}}dz=\Phi(a),$ ... will last more than $26$ months approximately equals $16\%$ is more than $15\%$ is less than $14\%$ is between $10\%$ and $15\%$
1 vote
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A permutation of $1,2, \dots, n$ is chosen at random. Then the probability that the numbers $1$ and $2$ appear as neighbour equals $\frac{1}{n}$ $\frac{2}{n}$ $\frac{1}{n-1}$ $\frac{1}{n-2}$
1 vote
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If a student copies his assignments from his friend he would get 80 marks. If he had done the assignments independently he would have scored 50 marks out of 100 and if the teacher finds he is cheating he will be penalized and will be given 0 marks. ... he copies 10 such assignments, what is the probability that he will lose more marks with copying than by doing his independent work independently?
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A carnival swing ride swings to the left with probability 0.4 and to the right with probability. If the ride stops after 10 swings, what is the probability that it is exactly at the place it started?
1 vote
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An airline operates a flight having 50 seats. As they expect some passenger to not show up, they overbook the flight by selling 51 tickets. The probability that an individual passenger will not show up is 0.01, independent of all other tourists. Each ticket costs Rs ... not available, the airline has to pay a compensation of Rs.1lakh to that passenger. What is the expected revenue of the airline?
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Suppose that the cdf of X is given by: F(a) ={ 0 for a < 0 1/5 for 0 ≤ a < 2 2/5 for 2 ≤ a < 4 1 for a ≥ 4. } Determine the pmf of X.
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Suppose that the running time for each process in milliseconds is an exponential random variable with parameter λ=1/20. If process P1 arrives immediately ahead of the process P2 in the running state, then the probability that process P2 will have to wait more than 20 milliseconds is _____________ . A 0.274 B 0.324 C 0.428 D 0.368 How to approach this. even not able to understand the question.
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15
Probability density function of a random variable X is distributed uniformly between 0 and 10 The probability that X lies between 2.5 to 7.5 and the mean square value of X are respectively. please give step by step answer in a detailed manner.
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In a lottery, 10 tickets are drawn at random out of 50 tickets numbered from 1 to 50. What is the expected value of the sum of numbers on the drawn tickets?
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A player tosses two fair coins. He wins rs 2 if 2 heads occur and rs 1 if 1 head occurs. On the other hand, he loses rs 3 if no heads occur. if the player plays 100 times.then the amount he wins______________(RS).
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This is an example in the book (A First Course in Probability by Sheldon Ross). A stick of length 1 is split at a point U that is uniformly distributed over (0,1). Determine the expected length of the piece that contains the point 0≤p≤1. So, My doubt here(see the blue mark) is according to proposition g(x) is Lp(U) , what is f(x) why f(x) is not multiplied. in calculating expected value.
1 vote
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A continuous random variable $x$ is distributed over the interval $[0,2]$ with probability density function $f(x) =ax^2 +bx$, where $a$ and $b$ are constants. If the mean of the distribution is $\frac{1}{2}$. Find the values of the constants $a$ and $b$. $a=2, b=- \frac{13}{6}$ $a= – \frac{15}{8}, b=3$ $a= – \frac{29}{6}, b=2$ $a=3, b= – \frac{7}{2}$
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Find the value of $\lambda$ such that function f(x) is valid probability density function $f(x)=\lambda (x-1)(2-x)$ for $1 \leq x \leq 2$ $=0$ otherwise My $\lambda$ is coming to be $- \frac{6}{5}$ Am I correct?
1 vote
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Suppose the random variable X has the probability distribution given below: X -2 -1 0 1 2 P(X=X) 0.25 0.20 0.15 0.35 0.05 Let $Y=(2*(X^2))+6$.The expected value E(Y) is: A) 9.5 B) 6. C )15.5. D )18
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23
Roll a die $n$ times and let $M$ be the number of times you roll $6$. Assume that $n$ is large. (c) How large should $n$ be so that the probability in(b) is larger than $0.99$?
1 vote
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24
Roll a die $n$ times and let $M$ be the number of times you roll $6$. Assume that $n$ is large. (b) Write down an approximation, in terms on $n$ and $\phi$, of the probability that $M$ differs from its expectation by less than $10$ %
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25
Roll a die $n$ times and let $M$ be the number of times you roll $6$. Assume that $n$ is large. (a) Compute the expectation $EM$.
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26
Toss a fair coin twice. You win $1$ dollar if at least one of the two tosses comes out heads. (b) Approximately how many times do you need to play so that you win at least $250$ dollar with probability at least $0.99$.
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27
Toss a fair coin twice. You win $1$ dollar if at least one of the two tosses comes out heads. (a) Assume that you play this game $300$ times. What is, approximately, the probability that you win at least $250$ dollar ?
1 vote
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28
After your complaint about their service, a representative of an insurance company promised to call you "between $7$ and $9$ this evening." Assume that this means that the time $T$ of the call is uniformly distributed in the specified interval. (c) Assume that you know in advance ... . Let $M$ be the amount of time of the show that you miss because of th call. Compute the expected value of $M$.
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29
After your complaint about their service, a representative of an insurance company promised to call you "between $7$ and $9$ this evening." Assume that this means that the time $T$ of the call is uniformly distributed in the specified interval (b) At $8.30$, the call still hasn't arrived. What is the probability that it arrives in the next $10$ minutes?
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30
After your complaint about their service, a representative of an insurance company promised to call you "between $7$ and $9$ this evening." Assume that this means that the time $T$ of the call is uniformly distributed in the specified interval. (a) Compute the probability that the call arrives between $8.30$ and $8.20$.