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.

learn more… | top users | synonyms

0
votes
0answers
4 views

Question on Stationary & Cointegration Test (Augmented Dickey Fuller & Engle Granger test)

I'm performing the stationary and cointegration test on stock prices. What I'm confused is 1) the difference between ADF stationary test and ADF cointegration test. 2) Also, in ADF stationary ...
0
votes
0answers
17 views

Hypothesis testing - Beta coefficient

Apologies if this is trivial, just linke me somewhere. I'm currently taking statistics 101, I can't wrap my head around the hypothesis testing of coefficients. As follows, the t-test reads $$T=\frac ...
0
votes
1answer
20 views

A random sample of size 5 is drawn from the pdf $f_{Y}(y) = 2y, 0\leq y \leq 1$. Calculate $P(Y_{(1)} < 0.6 < Y_{(5)})$. [on hold]

A random sample of size 5 is drawn from the pdf $f_{Y}(y) = 2y, 0\leq y \leq 1$. Calculate $P(Y_{(1)} < 0.6 < Y_{(5)})$. (Hint: Consider the complement.) Attempt: The pdf of the largest order ...
0
votes
0answers
11 views

Which model to be used for predictive analysis

I have a problem where i have been given set of data against month example Month | Data1 | Data2 1---------5--------5 2---------6--------7 Consider the data 1 be the temperature and data 2 be the ...
2
votes
1answer
17 views

Joint probability distribution

$Y_1$ and $Y_2$ are jointly distributed with density $f(y_1,y_2)=4y_2^2 \qquad 0 \leq y_1 \leq y_2 \leq 1$ Determine the following: $P$( max {$Y_1,Y_2$} $< 1/2$) $P(Y_1+Y_2 < 1/2)$ ...
0
votes
1answer
17 views

Given a pdf $f_{Y}(y)$ and $n$ random observations. Find probability that last observation will be the smallest number in all the sample?

Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the entire sample? attempt: ...
0
votes
0answers
9 views

Kth moment of the standard deviation from a normal population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the Kth moment of T about the origin, and state the condition for the existence ...
-1
votes
2answers
12 views

Probability of the highest order statistic below the population median.

What is the probability that the highest order statistic of a random sample of size n from any continuous distribution is below the median ( population median ) of that distribution.
0
votes
0answers
20 views

Presentation of 2 images in a random but counterbalanced way

Problem: For 18 trials randomly a ‘left’ labeled image or ‘right’ labeled image is shown. The first 9 trials should contain the opposite number of left images as the last 9 (a.k.a. counterbalance). ...
2
votes
2answers
20 views

do discrete probability distribution functions need a countable number of outcomes?

Everywhere I see on the internet they say that discrete probability distribution functions have a countable number of outcomes, and continuous have uncountable infinite number of outcomes. However if ...
0
votes
0answers
8 views

Unbiased Estimator and Variance in Polling

Say a pollster conducted m = 16 polls among people who voted in the 2010 presidential elections, and reports that 55% of the respondents would vote for John Smith. But the pollster did not report how ...
0
votes
0answers
8 views

Statistics matching answer check please

1.0.0968 < p-value < 0.1056 2.0.2119 < p-value < 0.2266 3.0.0278 < p-value < 0.0316 4.0.3422 < p-value < 0.3682 Possible Answers A. Ha: mu > 2.3, z* = -0.78 B. Ha: mu ...
0
votes
0answers
8 views

MSE in case of log-transformed dependent variable

Let's consider the following log-linear model: $log(Y_i) = \alpha + X_i\beta + \epsilon_i$ for i = 1, ..., N The fitted value is: $\widehat{log(Y)} = \hat{\alpha} + X\hat{\beta}$ Assuming ...
0
votes
1answer
15 views

Hypothesis testing, t procedures [on hold]

A realtor claims the mean income of households in a certain community is \$300,000. To check this claim, a potential resident samples 30 incomes of households in the community, and obtains a mean ...
0
votes
0answers
23 views

Critical values for hypothesis testing?

How do i determine the level of significance if i know the the critical values, and how do i do the opposite, on a normal distributed curve. I am asking because I am at the moment trying to calculate ...
2
votes
0answers
8 views

Non parametric estimators for noisy funcions

Suppose there is a function $f(a,b,c,\ldots)$ of $M$ variables (fixed numbers, not random variables). Add some Gaussian noise to this function: $$ g(a,b,c,\ldots) = f(a,b,c,\ldots) + ...
1
vote
0answers
23 views

A property of the hazard function of the normal distribution

I have a problem that I can't figure out. Define $\Gamma\left(x\right):=\frac{\phi(x)}{1-\Phi(x)}$, where $\phi(x)$, $\Phi(x)$ are the density respectively cumulative distribution function of the ...
0
votes
0answers
8 views

One-to-one correspondence between mean value and parameters

I am currently taking a course in statistics, and in this course we are considering linear models $\mu = X\beta$ where $\mu \in L$ and $L = col(X)$ is a linear subspace of $\mathbb{R}^n$, $X$ is the ...
0
votes
0answers
15 views

What is the limiting distribution of this Markov Chain?

Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$: Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the ...
-1
votes
0answers
17 views

CLT, mle, variance [on hold]

This is a practice problem that I don't know how to do. Let X_1,...,X_n be an i.i.d. sample from an exponential distribution with the density function. f(x/T) = (1/τ)*e^(-x/τ) 0<= x <= ...
0
votes
3answers
23 views

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement.

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement. Let $X$ be the number of yellow balls on the first two draws and $Y$ the number of ...
2
votes
0answers
11 views

Estimate of shared variance for n samples of x and y

I am performing a t-test on n different samples of both $X_1, X_2,...,X_k$ and $Y_1,Y_2,...,Y_k$. To begin with I want to assume that all 2*n samples have the same variance but that they do not have ...
1
vote
0answers
13 views

Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
1
vote
0answers
22 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(x=0)$.Here is my solution... $\mu= \frac {p(2) 2!}{p(1)}$. then how can find the mean..thanks
1
vote
0answers
6 views

What is the transformation that maps a Gaussian distribution to a Beta distribution?

Suppose X is a random variable with Gaussian distribution over domain $\mathbb{R} = (-\infty, +\infty)$, with PDF function $f_X$. And Y is a random variable with Beta distribution over domain ...
1
vote
1answer
21 views

Probability of Sample Variance Given Variance

I am trying to solve a problem that I have never seen before and cant seem to find a way to solve it so any help or tips would be appreciated! Here's the Problem: Suppose a considerable amount of ...
1
vote
0answers
23 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
1
vote
2answers
30 views

Variance of two functions

I have a problem where Var(X) is given as 8100, Var(Y) is given as 10,000. Var(X+Y) = 20,000. If X is increased by 500, Y is increased by 8%, such that the new formula is X+500 +(1.08)Y. How would I ...
0
votes
1answer
12 views

AP Probability problem on independence

This is a in-class practice problem. Suppose that the probability that a person has to park illegally and that he gets a parking ticket is 0.07. Last year Sam recorded data and found that because of ...
0
votes
1answer
35 views

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 } ...
1
vote
1answer
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?
0
votes
0answers
12 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 ...
1
vote
1answer
16 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 ...
1
vote
1answer
17 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 = ...
0
votes
1answer
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 ...
1
vote
1answer
18 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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
3
votes
0answers
21 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 ...
0
votes
1answer
12 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 ...
0
votes
1answer
12 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 ...
0
votes
0answers
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 ...
-1
votes
2answers
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 ...
0
votes
0answers
8 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? ...
0
votes
2answers
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 $x$?
0
votes
2answers
17 views

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: ...
1
vote
2answers
17 views

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.
2
votes
1answer
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 ...
0
votes
1answer
25 views

question on normal distribution [on hold]

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) ...
0
votes
0answers
4 views

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 ...