Given descriptions of  3 random variables, decide whether or not each is from a Poisson or Binomial distribution.

Choosing whether given random variables are qualitiative or quantitative.

$X \sim \operatorname{Binomial}(n,p)$. Find $P(X=a)$, $P(X \leq b)$, $E[X],\;\operatorname{Var}(X)$.

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Given a discrete random variable $X$ find the expectation of $1/X$ and $e^X$.

Arrivals given by exponential distribution, parameter $\theta$ and $Y$, sample mean on inter-arrival times. Find and calculate unbiased estimator for $\theta$.

Using a random sample from a population with given mean and variance, find the expectation and variance of three estimators of $\mu$. Unbiased, efficient?

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 $P(a \lt X \lt b)$.

Also find the expectation $\displaystyle \operatorname{E}[X]=\int_{-\infty}^{\infty}xf_X(x)\;dx$.

Given three linear combinations of four i.i.d. variables, find the expectation and variance of these estimators of the mean $\mu$. Which are unbiased and efficient?

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 $\overline{X}$.

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 $P(X \lt a)$.

$X \sim \operatorname{Binomial}(n,p)$. Find $P(X=a)$, $P(X \leq b)$, $E[X],\;\operatorname{Var}(X)$.

$W \sim \operatorname{Geometric}(p)$. Find $P(W=a)$, $P(b \le W \le c)$, $E[W]$, $\operatorname{Var}(W)$.

$Y \sim \operatorname{Poisson}(p)$. Find $P(Y=a)$, $P(Y \gt b)$, $E[Y],\;\operatorname{Var}(Y)$.

$X$ is a continuous uniform random variable defined on $[a,\;b]$. Find the PDF and CDF of $X$ and find  $P(X \ge c)$.

Choosing whether given random variables are qualitiative or quantitative.

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 $P(a \lt X \lt b)$.

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 $P(a \lt X \lt b)$.

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 $P(X \lt a)$.