# Tagged Questions

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### probability distributions ! what is the solution [closed]

among group of 500 students 30% are males 10% of males and 20% of females are left handed one student is selected at random what is the prob that the selected student is left handed ? help me please
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### Convergence in distribution and probability

Suppose ${X_{n}}$ is a sequence of non-negative random variables with cumulative distribution function given by $F_{X_{n}}(x) = 1 - 1/(1+nx)$ for $x\geq 0$. Examine if $\{X_{n}\}$ converges in ...
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### Finding the probability of a probability density function

Suppose that $f(x) = e^{−x}$ for $0 < x$. find $P(1 < X)$ I know typically we integrate $f(x)$ from $1$ to $\infty$ but in this case $x = 1$ is not included, how do I go about doing this? All ...
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### Joint distribution of range $(R=X_n-X_1)$ and mid-range $(V=\frac{1}{2}(X_1+X_n)$order statistics

Let $X_1,X_2, · · · , X_n$ be independent and identically distributed Uniform random variables on the interval (0, a) for a > 0, each having a density function $f(x) = \frac{1}{a}$, $0<x<a$. Let ...
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### Probability: Random Variables and Probability Distributions

1) The function: $F(x)=k(1-(1/2)^{[x]})$, $x > 0$ Is the distribution function for a discrete random variable X. Here, [x] denotes the integer part of x (i.e., the greatest integer less than or ...
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### $U$-Uniform$(0,2\pi)$, Z-Exp$(1)$, $U$ and $Z$ are independent. Then $\sqrt{2Z}\cos U$ and $\sqrt{2Z}\sin U$ are independent standard normal. [closed]

Given that $U$-Uniform$(0,2\pi)$, Z-Exp($1$), $U$ and $Z$ are independent. Show that $\sqrt{2Z}\cos U$ and $\sqrt{2Z}\sin U$ are independent standard normal variables. Thanks in advance for any ...
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### Exchangeable/Independent Bernoulli Distribution

Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having ...
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### Travelling from one destination to another

This is the problem : Manish has to travel from A to D changing buses at stops B and C enroute. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes ...
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### Probability of sample mean [closed]

A town has $500$ real estate agents. The mean value of the properties sold in a year by these agents is $\$800,000$and the standard deviation is$\$300,000$. A random sample of $100$ agents is ...
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### Apparently same probability questions with different answers.

I was reading A first course in probability by Sheldon Ross when and then I came up with this question. This is how he introduces the famous problem of points Independent trails, resulting in a ...
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### Probability and Expected profit

I really need help for this qn! You are asked to determine the profitability of a new line of sunglasses, which will retail for \$10. The fixed cost of setting up the line is \$2000. The total number ...
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### Flipping several biased coins

Assuming I'm flipping $M$ biased coins with different probability for heads $p_i, i=\{1,...,M\}$. What is the probability of having $k$ times head? Is there a distribution function known for this?
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### How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
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### Make the sum of random variables converge, while the sum of the variances diverges [closed]

Suppose $X_n$, $n=1,2,3,...$, are independent and $Var(X_n)$ is uniformly bounded by finite constant $C>0$. Construct $X_n$ such that $\sum_nX_n$ converges a.s., but $\sum_nVar(X_n)=\infty$.
Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
Let $X \sim N(0,1)$ and $G \sim Gamma(a)$. Why is $\frac{X}{G}$ t-distributed?