• 1.

  Ready to use
  Given a probability mass function $P(X=i)$ with outcomes $i \in \{0,1,2,\ldots 8\}$, find the expectation $E$ and $P(X \gt E)$.
• 2.

  draft
  Calculating simple probabilities using the exponential distribution.
• 3.

  draft
  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 $P(X \lt a)$.
• 4.

  draft
  3 Repeated integrals of the form $\int_a^b\;dx\;\int_c^{f(x)}g(x,y)\;dy$ where $g(x,y)$ is a polynomial in $x,\;y$ and $f(x)$ is a degree 0, 1 or 2 polynomial in $x$.
• 5.

  draft
  Repeated integral of the form: $\displaystyle I=\int_0^1\;dx\;\int_0^{x^{m-1}}mf(x^m+a)dy$  
• 6.

  Ready to use
  An experiment is performed twice, each with $5$ outcomes $x_i,\;y_i,\;i=1,\dots 5$ . Find mean and s.d. of their differences $y_i-x_i,\;i=1,\dots 5$.
• 7.

  draft
  Determine if the following describes a probability mass function. $P(X=x) = \frac{ax+b}{c},\;\;x \in S=\{n_1,\;n_2,\;n_3,\;n_4\}\subset \mathbb{R}$.
• 8.

  Ready to use
  Given subset $T \subset S$ of $m$ objects in $n$ find the probability of choosing without replacement $r\lt n-m$ from $S$ and not choosing any element in $T$.
• 9.

  Ready to use
  Rolling a pair of dice. Find probability that at least one die shows a given number.
• 10.

  draft
  $X \sim \operatorname{Binomial}(n,p)$. Find $P(X=a)$, $P(X \leq b)$, $E[X],\;\operatorname{Var}(X)$.
• 11.

  draft
  $W \sim \operatorname{Geometric}(p)$. Find $P(W=a)$, $P(b \le W \le c)$, $E[W]$, $\operatorname{Var}(W)$.
• 12.

  draft
  $Y \sim \operatorname{Poisson}(p)$. Find $P(Y=a)$, $P(Y \gt b)$, $E[Y],\;\operatorname{Var}(Y)$.
• 13.

  Ready to use
  A weighted coin with given $P(H),\;P(T)$ is tossed 3 times. Let $X$ be the random variable which denotes the longest string of consecutive heads that occur during these tosses. Find the Probability Mass Function (PMF), expectation and variance of $X$.
• 14.

  draft
  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.
• 15.

  Has some problems
  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.
• 16.

  draft
  $f(x,y)$ 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 $P(Y \le b|X \ge c)$
• 17.

  draft
  Given the PDF for $Y \sim \operatorname{Exp}(\lambda)$ find the CDF, $P(a \le Y \le b)$ and $\operatorname{E}[Y],\;\operatorname{Var}(Y)$
• 18.

  Ready to use
  $X$ is a continuous uniform random variable defined on $[a,\;b]$. Find the PDF and CDF of $X$ and find  $P(X \ge c)$.
• 19.

  draft
  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 $c \gt m$.
• 20.

  draft
  Given the parameters of a bivariate Normal distribution $(X,Y)$ find the parameters of the Normal Distributions: $aX,\;bY,\;cX+dY,\; Y|(X=f),\;X|(Y=g)$
• 21.

  Doesn't work
  Given normal distribution $\operatorname{N}(m,\sigma^2)$ find $P(a \lt X \lt b),\; a \lt m,\;b \gt m$ and also find the value of $X$ corresponding to a given percentile $p$%. 

There are no questions in this group.