Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

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Shortcut to finding $E(XY)$

The question says "Find $E(Y|X)$ and hence evaluate $E(Y)$ and $E(XY)$" The joint pdf is $$f_{X,Y}(x,y)=\begin{cases} 8xy, & \text{ for } 0< y< x < 1, \\0, & \text{ elsewhere } ...
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6 views

Poisson probability statisitics

In Poisson distribution, mean of babies born w/ defect is $1$ per month. What is the probability that exactly $12$ or exactly $14$ babies will be born w/ defect in $6$ months?
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14 views

identify nature of missingness for categorical variables

could you please give me some hints for identifying the nature of missingness for categorical variables' missing value? I mean, I gave a fast search on google scholar but I didn't find anything ...
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1answer
17 views

concept of one-tailed hypothesis testing

When we assume that the null hypothesis is true in one-tailed test for mean, we assume that the population mean is equal that value indicated in the hypotheses. Why do we not assume some other value ...
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1answer
19 views

Bivariate GBM - crosscovariance

I have troubles concerning a correlated bivariate GBM with identical drift and diffusion rates. Let $dX^i_t = \mu X^i_t dt + \sigma X^i_tdW^i_t$ and $E[dW_t ^idW^j_t] = \rho_{i,j}dt$ If $X_0^i = ...
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15 views

Inequality involving different diameter average

I have found an assertion in a scientific book (Hinds, Aerosol Technology, 2nd Edition, 1998, p. 83-84) that claims: Given the general form [here for grouped data] for the diameter of an average ...
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1answer
20 views

Is there an interpretation of the Beta Distribution?

There are cases in probability where one distribution has an "interpretation" in terms of another distribution: X ~ Gamma(k,1/m) for positive integer k, can be interpreted as the distribution of ...
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4 views

Recommendation needed in graph theory and statistics to be used in football predictions.

The following is a very simple model of what I am working on. I just need some advice since I don't have graph theory background. Suppose that A played at home against B and won by 3 goal ...
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28 views

Probability that the sum of random variables is less than some value

I would like to obtain the probability that the sum of random variables is smaller than some predefined value. Saying, $X_1, X_2, ..., X_n$ are independent random variables that come from the same ...
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32 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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1answer
17 views

How can I calculate the variance of a minimum of two random variables with different pdf and cdf?

I am trying to solve a problem where I need to find the variance of min (a,b). a is actually a function of a uniformly distributed r.v. while b is another r.v. with pdf and cdf as f and F. The support ...
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13 views

Bayesian Statistics: Estimators and Posterior Probability

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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7 views

Bayesian Statistics … Γ(α,β) Posterior Probability and Estimators

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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20 views

Probability of joint distribution

We were given some exercises to do to prepare for an upcoming quiz and there's one question that I'm struggling on. If $X ∼ N (μ = 10, σ^2 = 4)$ and $Y ∼ N (μ = 8, σ^2 = 16)$. Assume that X and Y ...
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9 views

Confidence interval question in Introduction to theory of statistics [on hold]

Problem: $X$ is a single observation from $$\theta \exp(-\theta x)I(0, \infty)(x)$$ where $\theta > 0$. a. $(X, 2X)$ is a confidence interval for $1/\theta$. What is the confidence coefficient? ...
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26 views

Finding expected value for random variable $X$ given a joint probability density function $f(x,y)$

I've been given $f(x,y) = 6y$ with boundaries $0 \leq y \leq x \leq 1$. How do I find the expected value of $x$?
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Want to find the maximum of an unnormalised density function.

Assume $\{Y_i\}$ are iid generated from a gamma distribution with shape $\alpha$ and rate $\beta$, $n$ is the number of $Y_i$. I have an unnormalised density function about $\alpha$ as follow: ...
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What is the probability that you will get at least one matching suit in 4 draws without replacement from a standard deck of cards?

I was wondering if someone could help me out with this one. I missed the lecture for this topic and am struggling to catch up. Could someone possibly explain this one to me? Thanks.
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29 views

Find $E(X)$ and $Var(X)$

In a box there are $30$ balls, $20$ are black and $10$ are red. Let $X$ be the number of red in a selection of two balls drawn without replacement then $$X=I_1 + I_2$$ where $I_1 = 1$ if red is drawn ...
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28 views

question on normal distribution [closed]

A mean of 25 mpg and a standard deviation of 5.8 mpg for highway driving. Assuming that a normal distribution can be applied. a) What gas mileage would put a car in 85th percentile for all cars? b) ...
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I need to compare 6 groups. I can use Kruskal-Walis test. Any other statistical tests I can use?

I have 6 groups of cell cultures I want to compare. I have the data on each of the groups, ex. the viability in each group (90%, 85%, 87%, 78%, 88%, 90%) and so on. How do I compare them? And how do ...
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24 views

Marginal Probability of Stochastic Process

I have a wide sense stationary stochastic process x(t)=asin(2πf0t)+bcos(2πf0t) where a & b are independent gaussian random variables. How can I find the Marginal probability of x(t)? I am ...
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21 views

Covariance Matrix Proof - Confusion with Cov(X,X) = Covmat(X)?

I have completed a proof regarding variance, covariance, and the covariance matrix. I think I have made a mistake regarding an assumtion. I need to show that $var(\{f\}) = a^TCovmat(\{x\})a$ Where ...
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25 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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55 views

Meaning of symbol $:=$

Can anyone tell me the meaning of this symbol $:=$ I couldn't find it online. It came up while I was studying joint probability of Gaussian random variables.
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1answer
25 views

Using CLT to calculate probability

The question I'm trying to answer says that the mean weight of luggage checked by a randomly selected passenger is 40 lb, and standard deviation is 10 lb. Luggage weights are independent. What is the ...
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1answer
25 views

chance to get number 4 in a fair dice in 100 throws [closed]

I am novice in Stats. If number four has appeared 64 times when a fair die was thrown 540 times? Using the above to get number 4 again in next throw (541st throw) Thanks Rodney
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11 views

mean & variance relating to bias

I am trying to prove something is biased, and I am given that its a SRS with mean μ = 0, and variance > 0. I think that since variance > 0, it should be biased, but I'm not sure. What does μ = 0 ...
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20 views

A composition of random variables, finding Expectation and Variance??? [closed]

The number of defects per yard $Y$ for a certain fabric is known to have a Poisson distribution with parameter $\lambda$. However, $\lambda$ itself is a random variable with probability density ...
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30 views

Finding MLE, MOM of a distribution

I'm stuck on a particular problem and I'm not quite sure what to do. The problem reads as such: Let $X_1, X_2, . . . , X_n$ be a random sample from a distribution with density $f(x) = ...
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Statistics linear modelling in R

Suppose I have a date set of the form: Test Subject/Sex/No. of mistakes made in the morning/ No. of mistakes made in the afternoon A / M / 2 / 5 B / F / 1 / 4 C / M / 3 / 5 D / F / 1 / 5 Suppose ...
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15 views

Probability equality related to simultaneous confidence intervals

Let $\tau_1,\ldots,\tau_q$ be a set of parameters and let $(a_1,b_1),\ldots,(a_q,b_q)$ be pairs of random variables where $a_i<b_i$ with probability 1 for all $i$. Show that $$\text{Pr}\left\{ ...
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1answer
48 views

Portfolio VaR with Copula?

Let the portfolio be given by: $$X=X_1+X_2$$ $(X_1,X_2)$ are dependent through a Copula function $C(u_1,u_2)$, such that the joint distribution is given by: $$F(x_1,x_2)=C(F(x_1),F(x_2))$$ What is ...
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Is there a such thing as a quasi-random shuffle?

I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce ...
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9 views

Auto-Thresholding PCA Eigenvalues

I'm applying a PCA on a dataset consisting of about 70k histograms with 153 bins each. So far everything is working fine except that I'm stuck on the decision which eigenvalues/vectors to throw away ...
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12 views

How to arrive at this conditional expectation $E[Y|A]= E[Y] + \frac{Cov(A,Y)}{Var(A)}(A-E[A])$

My professor arrived at the formula (1) but I cannot figure out how he got there. Is (1) from an identity after some manipulation? The setup is: X,Y are correlated, jointly gaussian random ...
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+50

Unsupervised learning algorithms to detect anomaly in waves.

I have a sample of graphs (more than 10000...). that look like in the image below: I am searching for an unsupervised learning algorithms that can help me to detect anomalous observations. Here ...
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How to check $H_0$ hypothesis using Pearson's criteria?

How to check hypothesis by using Pearson's criteria ( $\chi^2$ test), that $H_0:$ random variable $X$ is normally distributed given that $k=7$ (count of intervals) and $\alpha=0.1 $ (significance ...
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12 views

Cannot figure out autocavariance

The moving average model of order q has the form $$Y_t =β_0 +e_t +b_1e_{t−1} +b_2e_{t−2} +...+b_qe_{t−q}$$ where $e_t$ is a serially uncorrelated random variable with mean $0$ and variance $σ^2_e$. ...
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19 views

Expectation of higher order moment (Bayesian Stats)

I am taking a course in Bayesian statistics, which is off my field. In the lecture notes the instructor showed E[X^2n] = (2n-1) σ^2n and E[X^3 . Y] = E[Y^3 . X] = 3.ρ.σ^4 where σ = variance E = ...
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14 views

Time series regression

What if I have a stationary independent variable and 2 non-stationary dependent variables, and I want to run a regression on them, what model is the most appropriate?
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27 views

Statistics problem, minimum sample size for accurate estimate

A friend of mine recently had a problem at work, and needs to get an estimate for his boss, this is what he sent me: Assume I work at a food shipping company. A bunch of our crates of oranges ...
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Finding the probability of a type II error for a T distribution with unknown variance.

If is $\bar{X}$ the mean of a random sample of size $n$ from a normal distribution with mean $\mu$ and unknown standard deviation, then I know $\frac{\bar{X}-\mu}{S\sqrt{N}}$ has a T distribution with ...
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12 views

Posterior tail probability is absolutely continuous?

Suppose that the distribution of $X$ given $\theta$ is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$, for each value of $\theta$. Denote the conditional density with ...
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39 views

Formula the conditional probability of mables

I have a interesting question that need your help. I have two sets A and B. Set A have 10 marbles that numbered from 1 to 10. Set B have 6 marbles that numbered from 1 to 6. Randomly choose $g$ ...
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1answer
26 views

Maximum likelihood estimate in exponential distribution

A postal worker has a service time which is exponentially distributed with density $$f_{\lambda}(t)=\lambda \cdot e^{-\lambda t} , t\ge0$$ Given n observations $t_1, ... t_n$ find the maximum ...
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7 views

SPSS Statistical Analysis [closed]

Match the desired estimation on the left with the appropriate coefficient on the right. Desired Estimation mean difference between I.V.1 and I.V.3 mean of the D.V. for I.V.3 mean of the D.V. for ...
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1answer
16 views

Show that in this case $\rho \geq -\frac{1}{n-1}$

The bounds of correlation coefficient $\rho$ is shown to be $\pm 1$ in class. In many situations the bounds are sharper, i.e. they stay away from $+1$ or $-1$. Consider the random variables ...
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Show that $Y_1, Y_2$ are independent N(0,1) random variables.

Let $X_1, X_2$ be independent Uniform(0,1) random variables. Define $$Y_1=\,cos(2\pi X_1)\sqrt{-2\log(X_2)}$$ $$Y_2=\,sin(2\pi X_1)\sqrt{-2\log(X_2)}$$ Show that $Y_1, Y_2$ are independent N(0,1) ...
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1answer
25 views

Help with Linear Transformation of a multivariate normal

Given X ~ $N_2$ (μ, Σ)$ Find the Distribution of $$ \begin{pmatrix} X+Y \\ X-Y \end{pmatrix} $$ Show independence if $Var(X) = Var(Y)$ Attempt: Given proper of ...