All Questions
45 questions
22
votes
2
answers
2k
views
How do I analytically calculate variance of a recursive random variable?
Suppose I have a chest. When you open the chest, there is a 60% chance of getting a prize and a 40% chance of getting 2 more chests. Let $X$ be the number of prizes you get. What is its variance?
...
- 331
11
votes
1
answer
2k
views
Likelihood of my friend being able to guess skittle taste
I'm preparing for a data science interview, and here's a question I encountered during my preparation:
Your friend claims he can tell the five colors of skittles apart by
taste alone. The probability ...
user239903
8
votes
2
answers
13k
views
Expected value of maximum likelihood coin parameter estimate
Suppose I have a coin toss experiment in which I want to calculate the maximum likelihood estimate of the coin parameter $p$ when tossing the coin $n$ times. After calculating the derivative of the ...
- 83
8
votes
1
answer
12k
views
Variance and expectation of dot product
I am wondering what is the $E[\textbf{a}\cdot \textbf{b}]$ and $var[\textbf{a}\cdot \textbf{b}]$
where $\textbf{a}, \textbf{b}$ are independent random vectors. That is as a vector whose elements are ...
- 103
6
votes
2
answers
3k
views
Higher-dimensional version of variance
If $X$ is a real-valued random variable,
$$\mathbb{E}[X^2] - (\mathbb{E}[X])^2$$
is the variance of $X$.
Suppose now that $X$ is a random variable that takes values on $\mathbb{R}^n$. Consider the ...
- 6,738
5
votes
4
answers
6k
views
Practical meaning of expected value (mean value), variance and standard deviation?
I have a question about concepts:
Expected value (mean value) - $μ$
Variance - $σ^2$
Standard deviation - $σ$
What is the practical meaning of these common concepts of the probability theory and ...
- 379
5
votes
1
answer
541
views
Expected value and variance of moving a token on a cartesian plane based dice rolls
A fair four-sided die has its sides labeled U, D, L, and R, respectively. A token is placed at (0, 0) on the Cartesian plane and the die is then rolled repeatedly. After each roll, the token is moved ...
- 55
4
votes
2
answers
72
views
How $Var[e^{\frac{-1}{X+a}}]$ varies with $n$ where $X \sim Bin(n,p)$?
I have a binomial random variable $X \sim Bin(n,p)$. I am interested in the variance of a function $f(X)$ given by :
$f(X)=e^{\frac{-1}{X+a}}$. Here $a>0$.
Specifically, I would like to know how $...
- 224
4
votes
1
answer
1k
views
expected value of the dot product of normalized random vector and its mean
Suppose $U$ is a random vector satisfying $\mathbb E[U] = \mu$ and $\mathrm{var}(\|U\|_2) \le V$. Let $\bar{U} = U / \|U\|_2$ and $\bar\mu = \mu / \|\mu\|$. What is a lower bound on $\mathbb E[\bar U^\...
- 523
3
votes
3
answers
3k
views
Expectation of a square root of a sample mean
Let $X_i$ be iid exponential random variables. I want to calculate $Var\left(\sqrt{\bar{X}/6}\right).$
The idea I had to simplify this is expressing it as $$Var\left(\sqrt{\frac{\bar{X}}{6}}\right) =...
- 802
3
votes
2
answers
95
views
$X$ has distribution function $F(x) = e^{-e^{-x}}$. Justify that such a probability measure on $\mathbb{R}$ exists
How can I prove a probability measure exists? If $F(x) \rightarrow 1$ as $n \rightarrow +\infty$, does that mean $F(x)$ does exist? And how should I calculate $\mathbb{E}(F(X))$ and $Var(F(X))$?
- 31
3
votes
2
answers
3k
views
Finding the maximum and minimum variance of the sum of two Bernoulli events?
You are guessing the contents of two envelopes. Let $U_i$ be the event that you guess correctly. Your probability of guessing correctly for each envelope is $P(U_1) = P(U_2) = 3/4$. $U_1$ and $U_2$ ...
2
votes
3
answers
396
views
Expected value and variance of median
Suppose $Y|\Lambda\sim U(0,\lambda)$ with $\Lambda \sim U(0,1)$. If there is sample with size $n$ of $Y$ (To simplify, assume $n$ is odd, so $n=2m-1$). How do I calculate the expected value of median (...
- 23
2
votes
1
answer
553
views
Widgets and boxes problem: expectation and variance. Why is this wrong?
I'm taking the MITx: 6.041x Introduction to Probability - The Science of Uncertainty class to sharpen my probability skills. In one of the problems, the solution I came up with diverged from the ...
- 159
2
votes
1
answer
86
views
$E[XY]-E[X^2]-E[Y^2]$, is there any special property?
Given probability distributions of random variable $X,Y$, without any additional assumptions, is there any nice representation or properties of the combination $E[XY]-E[X^2]-E[Y^2]$? If not, is there ...
- 23
2
votes
1
answer
86
views
When would the variance for a probability distribution give the same result as the standard equation?
Variance equation for a probability distribution:
$$
\sigma^2 = \sum_{i=1}^{N}(x_i-\mu)^2P(X=x_i)
$$
Standard variance equation:
$$
\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2
$$
I understand that ...
- 123
2
votes
1
answer
5k
views
Variance of product of two random variables
I’m trying to calculate the variance of a function of two discrete independent functions.
The first function is $f(x)$ which has the property that:
$$\Bbb{P}(f(x)) =\begin{cases} 0.243 & \text{...
- 23
2
votes
2
answers
246
views
Expression of $E[(X-a)^3]$ as a function of $\operatorname{Var}(X)$ and/or $\sigma_x$
Just a question:
I would be able to express $\mathrm{E}\left[(X-a)^{3}\right]$ as a function of $\sigma_x$
and/or $\sigma_x^2$, with $a$ a constant (surely $\mathrm{E}\left[X\right]$ terms should ...
user226073
2
votes
1
answer
1k
views
Mean and Variance of dot product of 2 random vectors?
x and y are two vectors of dimension k.
Assume that the components of x and y are independent random variables with mean 0 and variance 1. What would be the mean and variance of their dot product, x · ...
- 21
2
votes
2
answers
911
views
MLE of variance is biased in a Gaussian distribution
Referring to: How to understand that MLE of variance is biased in a Gaussian distribution
at some point during calculation the formula of the sum of the expected value becomes a single expected value:...
- 33
2
votes
1
answer
56
views
A Doubt involving Variance Equation and Expectations
Consider the following,
$$
\begin{alignedat}{1}
\operatorname{Var}(X)&=E((X-E(X))^2)\\&=E(X^2)-(E(X))^2.
\end{alignedat}
$$
Since the expectation of a random variable is no longer random, let ...
- 143
2
votes
1
answer
191
views
what is the linear minimum mean squared estimator for y given x of the shaded region?
A 2D random point (x,y) is uniformly distributed on the shaded region of the figure.
What is the linear MMSE estimator for y, given x?
This is what I have so far:
Since it's a linear estimator, I ...
- 917
1
vote
1
answer
728
views
How to represent skewness(X) in terms of the expected value?
Let $X$ be the random variable. $E(X)$ is the expected value of $X$
Then
$Var(X)$ = $E(X^2)$ − $[E(X)]^2$
where $Var(X)$ is the variance of $X$
Then how to represent skewness(X) in terms of the ...
- 113
1
vote
1
answer
77
views
Finding variance?
$\newcommand{\Var}{\mathrm{Var}}$
Consider $Z_i$ as a binary random variable with $\mathrm{Pr}[Z_i = 1] = \pi$. Also, consider $Y_i$ as:
$Y_i|Z_i = 0 \sim \mathrm{Poisson} (\lambda_0) $
$Y_i|Z_i = 1 ...
- 159
1
vote
1
answer
564
views
Variance of expected value, is the formula right?
In this video and this video, I am seeing the variances of expected values calculated as this:
and this:
From which, I derived the formula:
$$\displaystyle\textrm{var}\big(\mathbf E[X\mid Y] \big) =...
- 233
1
vote
1
answer
126
views
Probability - expected value and variance
"A man is playing versus a machine in the following way: The machine chooses 2 numbers randomly from the set of numbers 1,2,3,4,5, where a number can be chosen twice (with replacement). If the ...
- 11
1
vote
1
answer
64
views
Name of the following minimization $E[(X - c)^2] = Var(X) + (E[X] - c)^2$ with $c = E[X]$
My professor proposed the below relationship as a property of the variance (he called $E[(X - c)^2]$ mean squared error):
$$
E[(X - c)^2] = Var(X) + (E[X] - c)^2
$$
and he said that, when $c = E[X]$, ...
- 221
1
vote
1
answer
63
views
Understanding covariance
I came across following problem:
A discrete random variable $P$ takes values $-3,-2,0,2,3$ with probability $0.2$. Let $Q=P^2$ be another random variable. What is covariance of $P$ and $Q$?
I solved ...
- 225
1
vote
1
answer
2k
views
Why is $E(u^2)=Var(y)$? (Binary Response Model)
I'm trying to show some results in binary response models, and a couple of the proofs use the "fact" that $E(u^2)=Var(y)$, but I can't see why this is.
The setup is that $y$ takes on the value $0$ or ...
- 97
1
vote
1
answer
57
views
Variance of $X + \alpha^\top Y$ where $X$ is a scalar random variable and $Y$ is a random vector [duplicate]
Consider a scalar random variable $X\in\mathbb{R}$, a vector random variable $Y\in\mathbb{R}^n$ and a constant (non-random) vector $\alpha\in\mathbb{R}^n$. I want to compute
$$
\mathbb{V}[X + \alpha^\...
- 667
1
vote
1
answer
41
views
Question relating to joint PDFs
Here are my questions:
Let $X$ ~ Unif$(0, 1)$, and $0<a<b<1$. Also, let
\begin{cases}
Y = 1 & \text{if $0<X<b$} \\
...
- 25
1
vote
1
answer
335
views
Computing $\mathbb{E}(S_n)$ and $\mathbb{V}(S_n)$ for Bernoulli data with a uniform probability parameter?
Take $U \sim \text{U}(0,1)$ as an underlying probability and generate $X_1,X_2,...,X_n \sim \text{Bern}(U)$ independent Bernoulli trials with this probability. The number of successes in the sample ...
- 13
1
vote
0
answers
86
views
How to find $\mathbb{E} \left[\frac{\bar{\mu}}{\bar{\sigma}^2}\right]$?
I asked the same question on math stacks: MathStacks:, and some user suggest to ask it here for better insight. So this question has found interest in many research problems, but there have been no ...
- 11
1
vote
0
answers
60
views
Show that if $Y$ is another random variable such that $E[X] = E[Y]$ and $V(X) = V(Y)$ then $P(Y \ge a) \le p$
Let $p \in (0,1)$ and $X$ be a random variable such that $P(X=a) = p, P(X=-b) = 1-p$
Show that if $Y$ is another random variable such that $E[X] = E[Y]$ and $V(X) = V(Y)$ then $P(Y \ge a) \le p$ and ...
- 71
1
vote
0
answers
79
views
CLT and 2 variables
Okay so there are 2 variables $D_i$ and $V_i$. Now $D= D_1 + D_2 + ... + D_N$ and $V = V_1 +.. +V_N$
Now I know the relationship is such that $E[D_i - a*V_i] = 0$ and
$Var[D_i - a*V_i] = E[D_i]$ ...
- 489
0
votes
2
answers
4k
views
Theoretical expected value and variance
Let $X$ be a random variable having expected value $\mu$ and variance $\sigma^2$. Find the Expected Value and Variance of $Y = \frac{X−\mu}{\sigma}$.
I would like to show some progress I've made so ...
- 9
0
votes
1
answer
32
views
Standard deviation of discrete variable
A start-up looking to get into the sleeveless shirt market is looking for \$10,000 from investors to get their company started. If you choose to invest this \$10,000, at the end of 5 years the company ...
- 1
0
votes
1
answer
116
views
The expected value and variance of E(-1X)? [closed]
This might be a stupid question, but how I can calculate the expected value $\operatorname{E}(-1X)$ and variance $\operatorname{Var}(-1X)$ for example in a case in which $X\sim N(100,0.1^2)$?
- 11
0
votes
1
answer
93
views
Probability - expected value
The random variable $X$ takes on values -2, 0 and 2 with probabilities 1/4, 1/2 and 1/4 respectively. Find $\text{E}(X)$ and $\text{Var}(X)$.
Till this part, it was easy enough.
Then the question ...
user218970
0
votes
0
answers
29
views
Expected value of a decreasing function of two random variables
My question is exactly equal to the question posted at Expected value of decreasing function of random variable versus expected value of random variable with just one extra assumption: the two random ...
- 1
0
votes
0
answers
25
views
Question regarding probability and maximum possible variance
I have the following question:
Suppose we have a set of 10 numbers (1, 2, ... , 10), each with a certain probability tagged to it.
Is it true that the highest possible variance is achieved when 1 and ...
- 31
0
votes
0
answers
63
views
Variance of $U= a \log (Z+b)-Z$ where $Z$ is the exponential random variable
Consider a random variable
\begin{align}
U= a \log (Z+b)-Z
\end{align}
where $a,b>0$ and $Z$ is an exponential random variable.
Question: Can we find the variance of $U$?
Things that I tried
...
- 205
0
votes
0
answers
108
views
Is the inverse of the sample variance uniformly integrable?
Let $X_1,X_2,\dots,X_n$ be a sample of $n$ independent and identically distributed observations of a continuous population random variable $X$. Define $Z_n$ to be the inverse of the sample variance:
$$...
- 893
0
votes
0
answers
77
views
Is the inverse of the sample variance integrable?
Is the inverse of the sample variance integrable? That is, does it hold that
$$
E\bigg[\bigg(\frac{1}{n}\sum_{i=1}^n X_i^2 - \overline{X}_n^2\bigg)^{-1}\ \bigg] < \infty.
$$
- 893
0
votes
0
answers
833
views
Joint Density and Covariance between Two Random Variables with the same Mean and Variance
This seems like a deceptively simple question, (and it perhaps is and I am missing something) but I could not find anything on this.
Q1)
Are there any general results / relationships to get the Joint ...
- 385