10 votes
Accepted

A complicated integral that Mathematica can't compute

Use characteristic functions and the fact that $\cos(x)=(e^{ix}+e^{-ix})/2$: $$E[\cos(X)]=\frac{1}{2}(E[e^{iX}]+E[e^{-i X}])=\frac{e^{-\frac{\sigma^2}{2}}}{2}(e^{i\mu}+e^{-i\mu})=e^{-\frac{\sigma^2}{2}...
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  • 8,751
4 votes

What is the distribution of the number of boys standing between the leftmost girl and the rightmost girl?

The total number of placements is $\binom{20}{10}$. The number of placements with $n$ boys between the outermost girls is $(11-n)\binom{8+n}8$, namely $11-n$ ways for the pair of outermost girls to be ...
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2 votes
Accepted

Application of the binomial distribution

Recall that $X_b=1$, if ball $b$ turned up red, and $0$ else. Then $p=P[X_b=1]= n^{-1/3}\ $ and $X:=\sum_{b=1}^n X_b$ is Bin$(n,p)$ distributed, with mean $\mu=np=n^{2/3}$. The relevant ...
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  • 13.4k
2 votes
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Expected value of $X^3$

If $X\sim f_X(x)$, then $$ \begin{aligned} \mathbf{E}\left[X\right] &= \int_{-\infty}^{+\infty}xf_X(x)dx \\ \mathbf{E}\left[g(X)\right] &= \int_{-\infty}^{+\infty}g(x)f_X(x)dx, \text{for some ...
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  • 1,285
2 votes
Accepted

Number of vowels in the first 5 letters of a random arrangement

I think I'll go for this, given I had a helpful comment first up. I'll call Elizabeth Taylor as Liz Taylor or Liz below, I hope she does not mind. Our idea is to think about the individual components ...
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2 votes
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If a value lies at EXACTLY two standard deviations from the mean, is it "usual data" or "unusual data"?

I have a feeling that this question solely depends on your definition, and there's no underlying mathematical principle that would tell you a definite answer. Just consult your textbook on the ...
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  • 36
2 votes
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Show $\mu \ge x (1 - F(x))$ for $x \in [0,1]$

$\mu=\int_0^{1} tdF(t) \geq \int_x^{1} tdF(t) \geq \int_x^{1} xdF(t)=x\int_x^{1}dF(t)=x(1-F(x)$.
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  • 2,660
2 votes
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Related to order statistics?

Using the assumptions we can develop the equation you provided. Let $T_1 = \min_{i}{Z_i}$, and let's find the PDF of $T_1$, by first deriving the CDF of $T_1$: $$P(T_1 \leq t) = P(\min_{i}{Z_i} \leq t)...
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  • 102
1 vote

Probability in terms of CDF and PDF

Well, yes but that's quite trivial, since the first factor inside the integral is constant and what you get is just $1$ times it. What you could do, is put the indicator of the event that's inside the ...
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  • 6,934
1 vote

What is the distribution of the number of boys standing between the leftmost girl and the rightmost girl?

Between the leftmost girl and the rightmost girl are always 8 girls plus $0$ to $10$ boys. Now: If there is no boy between them, the group of $\color{red}{10}$ girls may be in 11 different ...
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  • 2,830
1 vote

Expected value of $X^3$

I know that mistake is somewhere when finding $f_{X^3}(x)$ Take some uniformly spread sample points, say $\{0.00,0.25,0.50,0.75,1.00\}$, and examine their cubes, $\{0, 0.015625, 0.125, 0.421875, 1\}$....
1 vote
Accepted

Lower bound for Probability of Maximum of Normal Variables

Let $Y = \max_{1\le i \le n} X_i$. The event that $Y\ge c\sqrt{\log(n)}$ is the complement of the event $\bigcap_{i=1}^{n} \left\{X_i < c\sqrt{\log(n)} \right\}$. Since the $X_i$ are iid, it ...
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1 vote
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How to do change of variables in multivariate normal distribution?

Lets rewrite your variables, $$ q = \begin{bmatrix} x\\ y\\\end{bmatrix}\\ \mu = \begin{bmatrix} \mu_x\\ \mu_y\\\end{bmatrix}\\ \Sigma = \begin{bmatrix} \Sigma_{xx} & \Sigma_{xy}\\ \Sigma_{yx} &...
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1 vote

Related to order statistics?

Lack of independence makes this too complicated. Independent but not identically distributed is still possible to analyse: The CDF of the minimum is $$F_{Z_{(1)}}(x) = 1- \prod\limits_i (1-F_{Z_i}(z))...
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  • 143k
1 vote
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why selecting objects one by one without replacement and selecting objects at the same time give same probability?

Suppose I shuffle a deck of cards and deal you 5 cards, but you were looking away at the time and didn't see whether I took all 5 off the top at once or dealt 5 individually. Could this have possibly ...
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  • 17.9k
1 vote
Accepted

Name for half life where X axis is not time

It is called the "decay constant."
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