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Calculate probabilities from a normal distribution,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given a random variable $X$ normally distributed as $\operatorname{N}(m,\sigma^2)$ find probabilities $P(X \gt a),\; a \gt m;\;\;P(X \lt b),\;b \lt m$ .
Find expectation, variance and probability sample mean in range for normal distribution, ,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Normal distribution $X \sim N(\mu,\sigma^2)$ given. Find $P(a \lt X \lt b)$ . Find expectation, variance, $P(c \lt \overline{X} \lt d)$ for sample mean $\overline{X}$ .
Use piecewise CDF to find probabilities at given points, and the expectation.
Has some problems
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Given a piecewise CDF $F_X(b)$ which is discontinuous at several points, find the probabilities at those points and also find the value of $F_X(b)$ at a continuous point and the expectation.

This cdf is a step function and is therefore the cdf of a discrete random variable. This should be stated somewhere in the statement or the solution. Apart from this the question is correct.
Sampling with and without replaement - binomial, hypergeometric, and Poisson approximation,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Three parts. A sample of size $n$ is taken from $N$ where $k$ of the items are known to be defective and the task is to find the probability that more than $m$ defectives are in the sample. First part is sampling with replacement (binomial), second is sampling without replacement, (hypergeometric) and the last part uses the Poisson approximation to the first part.
Calculate probability, CDF, expected value and variance of binomial distribution,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$X \sim \operatorname{Binomial}(n,p)$ . Find $P(X=a)$ , $P(X \leq b)$ , $E[X],\;\operatorname{Var}(X)$ .
Conditional probability in normal distribution,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given a normal distribution $X \sim N(m,\sigma^2)$ find $P(X \lt a),\; a \lt m$ and the conditional probability $P(X \gt b | X \gt c)$ where $b \lt m$ and $c \gt m$ .
PDF and CDF of continuous uniform random variable,
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$X$ is a continuous uniform random variable defined on $[a,\;b]$ . Find the PDF and CDF of $X$ and find $P(X \ge c)$ .
Construct PDF and find CDF,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

The random variable $X$ has a PDF which involves a parameter $k$ . Find the value of $k$ . Find the distribution function $F_X(x)$ and $P(X \lt a)$ .
Bivariate continuous distribution - marginals and conditional probability,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$f(x,y)$ is the PDF of a bivariate distribution $(X,Y)$ on a given rectangular region in $\mathbb{R}^2$ . Write down the limits of the integrations needed to find $P(X \ge a)$ , the marginal distributions $f_X(x),\;f_Y(y)$ and the conditional probability $P(Y \le b|X \ge c)$
Calculate probability, expected value and variance of a Poisson distribution,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$Y \sim \operatorname{Poisson}(p)$ . Find $P(Y=a)$ , $P(Y \gt b)$ , $E[Y],\;\operatorname{Var}(Y)$ .
Is the given function a probability mass function?, , ,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Determine if the following describes a probability mass function.

$P(X=x) = \frac{ax+b}{c},\;\;x \in S=\{n_1,\;n_2,\;n_3,\;n_4\}\subset \mathbb{R}$ .
Find parameters of normal distributions from bivariate distribution,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given the parameters of a bivariate Normal distribution $(X,Y)$ find the parameters of the Normal Distributions: $aX,\;bY,\;cX+dY,\; Y|(X=f),\;X|(Y=g)$
Calculate probability, expected value and variance of a geometric distribution,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$W \sim \operatorname{Geometric}(p)$ . Find $P(W=a)$ , $P(b \le W \le c)$ , $E[W]$ , $\operatorname{Var}(W)$ .
Calculate probabilities using the exponential distribution,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Calculating simple probabilities using the exponential distribution.
Find CDF of given exponential distribution, and expectation and variance,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given the PDF for $Y \sim \operatorname{Exp}(\lambda)$ find the CDF, $P(a \le Y \le b)$ and $\operatorname{E}[Y],\;\operatorname{Var}(Y)$
Calculate probabilities from binomial distribution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Application of the binomial distribution given probabilities of success of an event.

Finding probabilities using the binomial distribution.
Ricardo's copy of IS2.2
Ready to use
Question in Ricardo's workspace by Ricardo Monge

The random variable $X$ has a PDF which involves a parameter $c$ . Find the value of $c$ . Find the distribution function $F_X(x)$ and $P(a \lt X \lt b)$ .
IP3.2
Ready to use
Question in Bill's workspace by Bill Foster

Determine if the following describes a probability mass function.

$P(X=x) = \frac{ax+b}{c},\;\;x \in S=\{n_1,\;n_2,\;n_3,\;n_4\}\subset \mathbb{R}$ .
IP3.1
Ready to use
Question in Bill's workspace by Bill Foster

Given a probability mass function $P(X=i)$ with outcomes $i \in \{0,1,2,\ldots 8\}$ , find the expectation $E$ and $P(X \gt E)$ .
IS2.1
Ready to use
Question in Bill's workspace by Bill Foster

The random variable $X$ has a PDF which involves a parameter $k$ . Find the value of $k$ . Find the distribution function $F_X(x)$ and $P(X \lt a)$ .
IS2.2
Ready to use
Question in Bill's workspace by Bill Foster

The random variable $X$ has a PDF which involves a parameter $c$ . Find the value of $c$ . Find the distribution function $F_X(x)$ and $P(a \lt X \lt b)$ .

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