5639489
Description
Flashcards by Daniel Cox, updated more than 1 year ago
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Created by Daniel Cox
about 8 years ago
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| Question | Answer |
| What does it mean if events A and B are mutually exclusive? Also, \(P(A\cap B)=?\) | Events A and B cannot happen at the same time. \[P(A\cap B)=0\] |
| What does it mean if events A and B are independent? Also, \(P(A\cap B)=?\) | If A happens, this does not affect the probability of B happening (and vice versa). \[P(A\cap B)=P(A) \times P(B)\] |
| \[P(A|B)=?\] (there is a rearranged version of this given in the formulae book) | \[P(A|B)=\frac{P(A\cap B)}{P(B)}\] |
| If events A and B are independent, then \(P(A|B)=?\) | \[P(A|B)=P(A)\] |
| If events A and B are independent, then \(P(B|A)=?\) | \[P(B|A)=P(B)\] |
| The addition law for events A and B is \[P(A\cup B)=?\] (given in formulae book) | \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\] |
| \[P(A')=?\] | \[P(A')=1-P(A)\] \(A'\) is called the complement of \(A\) and \(P(A')\) is the probability of \(A\) not happening |
| For events A and B that are NOT independent, \[P(A\cap B)=?\] | \[\begin{align*} P(A\cap B)&=P(A)\times P(B|A)\\ & =P(B)\times P(A|B) \end{align*}\] |
| Describe this shaded area using set notation | \[A\cap B'\] or \[B'\cap A\] |
| What is a sample space? | The set of all the possible outcomes of a random experiment |
| How many unordered samples of size \(r\) can be taken from a collection of \(n\) objects? | \[nCr=\binom{n}{r}=\frac{n!}{r!(n-r)!}\] make sure you know how to get your calculator to do this |
| For any discrete random variable \(X\),\[\text{E}(aX + b) = ?\] | \[\text{E}(aX + b) = a\text{E}(X) + b\] |
| For any discrete random variable \(X\),\[\text{Var}(aX + b) = ?\] | \[\text{Var}(aX + b) = a^2 \text{Var}(X)\] |
| For a discrete random variable \(X\) taking values \(x_i\) with probabilities \(p_i\), \[\text{E}(X)=?\] (given in formulae book) | \[\text{E}(X)=\sum x_i p_i \] |
| For a discrete random variable \(X\) taking values \(x_i\) with probabilities \(p_i\), \[\text{Var}(X)=?\] (given in formulae book) | \[\begin{align*} \text{Var}(X)&=\sum x_i^2 p_i -\mu^2\\ &=\text{E}(X^2)-(\text{E}(X))^2 \end{align*}\] |
| Describe this shaded area using set notation | \[A'\cap B\] or \[B\cap A'\] |
| Give the formula for the expected value of a function \(g(X)\) of a discrete random variable (given in formulae book) | \[E[g(X)]=\sum g(x) P(X=x)\] |
| \[X \sim B(n,p)\] \(\text{E}(X)=?\) (given in formulae book) | For the binomial distribution \(X \sim B(n,p)\), \(\text{E}(X)=np\) |
| \[X \sim B(n,p)\] \(\text{Var}(X)=?\) (given in formulae book) | For the binomial distribution \(X \sim B(n,p)\), \(\text{Var}(X)=npq=np(1-p)\) |
| \[X \sim Po(\lambda)\] \(\text{E}(X)=?\) (given in formulae book) | For the Poisson distribution \(X \sim Po(\lambda)\), \(\text{E}(X)=\lambda\) |
| \[X \sim Po(\lambda)\] \(\text{Var}(X)=?\) (given in formulae book) | For the Poisson distribution \(X \sim Po(\lambda)\), \(\text{Var}(X)=\lambda\) |
| Describe this shaded area using set notation | \[A \cup B\] or \[B \cup A\] |
| How would you use the Binomial or Poisson tables to find \(P(X=n)\)? | \[P(X=n)=P(X\leq n)-P(X\leq n-1)\] |
| How would you use the Binomial or Poisson tables to find \(P(X>n)\)? | \[P(X>n)=1-P(X\leq n)\] |
| How would you use the Binomial or Poisson tables to find \(P(X\geq n)\)? | \[P(X\geq n)=1-P(X\leq n-1)\] |
| How would you use the Binomial or Poisson tables to find \(P(X<n)\)? | \[P(X<n)=P(X\leq n-1)\] |
| For a continuous probability distribution, how are \(f(x)\) and \(F(x)\) related? | \[f(x)=F'(x)\] \[F(x)=P(X\leq x)=\int _{-\infty} ^x f(t) \, \text{d}t\] |
| If \(q\) is the lower quartile of a continuous random variable \(X\) with cumulative distribution function \(F\), then \[F(q)=?\] | \[F(q)=P(X\leq q)=0.25\] |
| Describe this shaded area using set notation | \[A \cap B\] |
| If \(m\) is the median of a continuous random variable \(X\) with cumulative distribution function \(F\), then \[F(m)=?\] | \[F(m)=P(X\leq m)=0.5\] |
| If \(Q\) is the upper quartile of a continuous random variable \(X\) with cumulative distribution function \(F\), then \[F(Q)=?\] | \[F(Q)=P(X\leq Q)=0.75\] |
| Give the formula for the expected value of a function \(g(X)\) of a continuous random variable (given in formulae book) | \[E[g(X)]=\int g(x) f(x) \, \text{d}x\] |
| For a binomial distribution \(X\sim B(n,p)\), what is the formula for \(P(X=x)\)? (given in formulae book) | \[\binom{n}{x}p^x(1-p)^{n-x}\] |
| For a Poisson distribution \(X\sim Po(\lambda)\), what is the formula for \(P(X=x)\)? (given in formulae book) | \[e^{-\lambda}\frac{\lambda^x}{x!}\] |
| Describe this shaded area using set notation in two ways | \[A'\cap B'\] or \[(A\cup B)'\] |
| How is variance related to standard deviation? | \[\text{variance}=(\text{stand. dev.})^2\] OR \[\text{stand. dev.}=\sqrt{\text{Variance}}\] |
| \[X\sim B(n,p)\] What values can \(X\) take? | \[0, 1, 2, ..., n\] |
| \[X\sim Po(\lambda)\] What values can \(X\) take? | \[0, 1, 2, ...\] (There is no maximum) |
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