5
votes
Why is $E[X^{2}] \ge E[X]^{2}$?
Note that expanding and simplifying the definition of variance gives
$$E[(X-E[X])^2]=E[X^2]-(E[X])^2.$$
Conclude by recognizing that for real-valued random variable $X$, the LHS is the mean of $(X-E[X]...
- 11.4k
2
votes
Probability and Random Variables.
The event $\{|X|<1.5\}$ is the same as the event $\{-1.5 < X<1.5\}$, therefore
$$
P(|X|<1.5) = P(-1.5 < X<1.5).$$
This explains where the $-1.5$ comes from.
The example is performing ...
- 35.9k
2
votes
Probability and Random Variables.
Note that $|X|<1.5\iff -1.5<X<1.5.$ Also note for real-valued random variable $X$ having density $f$,
$$\int_Af(x)dx=P(X\in A)=E[1_{X\in A}]=\int_{\mathbb{R}} 1_{X\in A}f(x)dx.$$
Your ...
- 11.4k
2
votes
If $\mathbb EX=0$ and $\mathbb E|X| < \infty $ implies finite Variance?
More generally, if the $k$th moment is finite, the moment of order $k+1$ need not be finite. For instance, the $t$ distribution with $k>1$ degrees of freedom has moments of order $1,2,...,k-1$ ...
- 11.4k
2
votes
Accepted
Triviality of the Chernoff bound for $t$ less than the expectation of the random variable
By Jensen's inequality, $\psi_X(\lambda)\geqslant \log e^{\lambda\mathbb E[X]}=\lambda\mathbb E[X]$ and if $\mathbb E[X]\geqslant t$, then $\psi^*(t)\leqslant 0$. Moreover, taking $\lambda=0$, we get ...
- 165k
1
vote
Intuitive/heuristic explanation of Polya's urn
Here is a simple explanation. Imagine you have a deck of $n+1$ cards, consisting of cards numbered $1$ to $n$ and a joker. We start with a stack consisting of only the joker. There are then $n$ steps, ...
- 64.9k
1
vote
Accepted
Binomial distribution probability (forming an equality)
Your approach is correct.
The probability that Sally does not play football in a given week is $1 - 0.29 = 0.71$. Therefore, the probability that she does not play football in a period of $n$ weeks ...
- 72k
1
vote
Accepted
Conditional Probability from textbook example unclear
Using your notation:
$P(B|A) = P(A,B) / P(A)$
$= P(A,B) / (P(A,B)+P(A,B’))$
where $B’$ is the event that the sample was not chosen from line 1.
One test failed while two succeeded. The failure might ...
- 422
1
vote
Probabilistic interpretation of Fourier Transform
The characteristic function of a real-valued random variable $X$ is given by
$$\varphi_X (t):=E[e^{itX}],$$
which is an object that completely characterizes the distribution of $X$. If $X$ admits a ...
- 11.4k
1
vote
Is posterior probability affected by a hidden observer?
Probability is relative to what we know ... what information we have.
OK, so with this game, everyone has a probability of $\frac{1}{10}$ of winning. To be precise: If we know that 10 cards are being ...
- 95k
1
vote
How to determine number of entries in buckets of a perfect normal distribution?
It won't be a perfect normal distribution since percentages (even if one allows tenths of a percent) are discrete, so such buckets would just be a close approximation of the normal distribution, which ...
- 106
1
vote
Why do singleton events imply sets in multiplicative but not in additive probability?
What about your own argument:
$$
\sum_{y \in Y} P(y \mid M) \leq \sum_{y \in Y} \left(aP(y \mid N)+b\right) \Longrightarrow ?
$$
how does the second sum distribute over summands compared to factors?
- 18k
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