# Tagged Questions

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### Truncated Poisson Asymptotics

This is a homework problem. I have figured out part (a) but I need help with part (b). I include part (a) for completion. Suppose $X_1,\ldots,X_n$ are iid Poisson random variables. Furthermore, let ...
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### I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process and I find mean and covariance functions

Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. Find its mean and covariance functions. thanks .
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### Bivariate conditional probability confidence

I have two random variables following normal distributions, $X\sim N(\mu_{x},\sigma_{x})$ and $Y\sim N(\mu_{y},\sigma_{y})$ I know the covariance is $\operatorname{Cov}(X,Y) = c$. And I believe ...
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### Two normal distributions

I will mark this as homework, even though it's just general interest. The scenario is a little silly, because it's just a real world case for a theoretical problem I've been thinking about. A man ...
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### Question about a marginal distribution

If I observe the following: $X \sim N(\mu_x,\sigma^2_x)$ $Y|X=x \sim N(x,\sigma^2_y)$ My objective is to calculate the marginal distribution of $Y$. (Since the variance term does not address some ...
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### Find the moment generating function

Find the moment generating function of $W$, when $W=X+2Y+4Z$. $~X,~Y,~Z$ are independent normal distributions $\mathcal N(1,4),~ \mathcal N(2,9) \text{ and }\mathcal N(3,16)$.
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### Simple homework question about normally distributed variables

The question states: Consider a set of random variables $X_i$, where $i=1,...n$. Each $X_i$ is normally distributed with mean $0$ and variance $1$, i.e. $X_i$ are $\mathcal N(0,1)$. What is ...
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### How to correctly model noise?

Assume a linear mixing model $x = As$, where $x = (x_{0}, ..., x_{n})^T$ are linear mixtures of $s = (s_{0}, ..., s_{n})^T$, and $A$ is the mixing matrix. Now, if I introduce additive noise to this ...
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### Questions about the sampling distribution of the sample mean

(a) Let $X_1,X_2,\cdots,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Show that $E[\bar{X}]=\mu$ and $V[\bar{X}]=\frac{\sigma^2}{n}$ What is the ...
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### Hypothesis testing of normal distribution, known mean unknown variance

I've been working on review problems, and this one has me completely stumped. Let $X_1 ... X_{10}$ be a random sample from a $N(3,\sigma^2)$ distribution, where $\sigma^2$ is unknown. Using the ...
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### Normal approximation to binomial

What do I do when the normal approximation is not valid? Here's the question I'm trying to answer: A student guesses all 15 answers on a multiple-choice test. There are 5 choices for each of the ...
### Find the normal probability $P(|X|<1)$ using z values?
If I have a random variable that follows a normal distribution i.e. $X \sim N(-3,4)$ and I want to calculate $P(|X| <1)$ how would I go about doing so, using the z values? Seeing as it's the ...