Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [symmetry]

94 questions
31
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4 answers

Does mean=mode imply a symmetric distribution?

I know this question has been asked with the case mean=median, but I did not find anything related to mean=mode. If the mode equals the mean, can I always conclude this is a symmetric distribution? Will I be forced to know also the median for this…
tzipy
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Does mean = median imply that a unimodal distribution is symmetric?

For a unimodal distribution, if mean = median then is it sufficient to say that distribution is symmetric? Wikipedia says in relationship between mean and median: "If the distribution is symmetric then the mean is equal to the median and the…
kaka
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2 answers

What is the definition of a symmetric distribution?

What's the definition of a symmetric distribution? Someone told me that a random variable $X$ came from a symmetric distribution if and only if $X$ and $-X$ has the same distribution. But I think this definition is partly true. Because I can present…
shijing SI
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12
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6 answers

Example distribution where 74% of probability is above the mean

Watching Why You Should Want Driverless Cars On Roads Now, at 8:14 Derek Muller claims: Surveys show 74 % of people believe they are above average drivers. This claim motivates my question, but some clarification is needed. I am not asking for…
DifferentialPleiometry
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9
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1 answer

If I know the density I'm estimating is symmetric about 0, how to impose this restriction in my kernel density estimator?

Suppose I'm interested in estimating the unknown smooth density of $X$ denoted by $f(\cdot)$ using data $\{X_i\}_{i=1}^{n}$. Suppose I also know that $f(\cdot)$ is symmetric about 0 in the sense that $f(-x)=f(x)$ for any $x$ in the support. My…
T34driver
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8
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2 answers

Testing for symmetric distributions

Suppose we have $n$ samples $s_1,...,s_n$ from an unknown real-valued distribution $D$. We are interested in a statistic to test if $D$ is symmetric around zero. (In my application, $n$ is only about 50, so I'm interested in the non-asymptotic…
Bill Bradley
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8
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1 answer

Why are mean and median not equal for asymmetric distributions?

My reasoning is as follows: the p.d.f. is divided by the mean (expected value) into two parts, for which the areas under the p.d.f. curve are equal, hence the probabilities that random variable takes a value less then or equal to the mean are 0.5,…
Marie
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7
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2 answers

Prove that a distribution is symmetric using moments

Given, a random variable X whose mean , variance and fourth central moment are 0, 2 and 4 respectively. Now, how do I prove that (1) third moment is 0 (2) distribute is symmetric about 0 and (3) X is bounded. With the above information the only…
Harry
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2 answers

$X$, $Y$ independent identically distributed. Are there counterexamples to symmetry of $X-Y$?

That $X-Y$ should be symmetrically distributed for iid $X,Y$ is obvious simply by interchanging the roles of $X$ and $Y$ -- informally we might argue Let $Z=X-Y$ have distribution $F$. The roles of which observation was called $X$ and which $Y$ is…
Glen_b
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6
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2 answers

Is there an estimator for the symmetry of a bimodal distribution?

I would like to know how I can measure the degree of symmetry of a bimodal distribution. Is there any a criterion like, for example skewness, in the case of unimodal distributions?
alexi
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3 answers

Distributions similar to Normal distribution

Can anyone give me examples of distributions symmetric around the mean but that are more densely concentrated around the mean than the Normal distribution for the same mean and variance? Thanks
jpcgandre
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5
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1 answer

What can we say about distributions of random variables $X$ such that $X$ and its inverse $1/X$ have the same distribution?

What can we say about random variables such that it and its inverse have the same distribution? One example is Cauchy distributed random variables, easily proved via the fact that if $X, Y$ are IID standard normal then both $X/Y$ and $Y/X$ are…
kjetil b halvorsen
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5
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Identically distributed vs P(X > Y) = P(Y > X)

I've two related propositions which seem correct intuitively, but I struggle to prove them properly. Question 1 Prove or disprove: If $X$ and $Y$ are independent and have identical marginal distributions, then $\mathbb{P} (Y > X) = \mathbb{P} (X >…
farmer
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5
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Algorithms for data symmetrization

There are statistical methods (e.g. by Box-Cox or Yeo-Johnson, see references below) to automatically bring data vectors as close as possible to symmetry/normality using optimal power transformations. Sometimes, the mentioned methods fail badly. And…
Michael M
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1 answer

symmetric r.v. raised to an odd power

My prof claims that raising a symmetric r.v., like N(0,1), to an odd power gives a distribution with expectation 0. What's the best way to see this?
Dan Schwartz
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