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Flashcards by Daniel Cox, updated more than 1 year ago
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Created by Daniel Cox
almost 8 years ago
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Copied by Daniel Cox
almost 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 |
| 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 \[B\cap A'\] |
| Describe this shaded area using set notation | \[A \cup B\] or \[B \cup A\] |
| Describe this shaded area using set notation | \[A \cap B\] |
| 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 \[\text{stand. dev.}=\sqrt{\text{Variance}}\] |
| The cumulative distribution function for a discrete random variable: \[F(x_0)=P(?)\] | \[F(x_0)=P(X\leq x_0)\] |
| If \(X\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma^2\), how do you transform it to the \(Z\) distribution? | \[Z=\frac{X-\mu}{\sigma}\] |
| \[\text{Interquartile range (IQR)}=?\] | \[\text{IQR}=Q_3-Q_1\] where \(Q_3\) is the upper quartile and \(Q_1\) the lower quartile |
| What is the formula for the mean of a set of data? | \[\bar{x} =\frac{\sum x}{n} \text{ or }\frac{\sum fx}{\sum f}\] |
| What is the underlying feature associated with each of the bars in a histogram? | Area is proportional to frequency |
| How do you find the range of a set of data? | \[\text{range}=\text{highest value}-\text{lowest value}\] |
| What is the formula for the standard deviation of a set of data? | \[\sigma =\sqrt{\frac{\sum x^2}{n}-(\bar{x})^2}\] OR \[\sigma =\sqrt{\frac{\sum fx^2}{\sum f}-(\bar{x})^2}\] |
| What is a continuous variable? | A variable that can take any value in a given range |
| What is a discrete variable? | A variable that can take only specific values in a given range |
| What is \(r\) (the product moment correlation coefficient) a measure of? | \(r\) is a measure of linear correlation |
| \(r\) is the product moment correlation coefficient \[\begin{align*} r=1 &\Rightarrow ?\\ r=-1 &\Rightarrow ?\\ r=0 &\Rightarrow ?\\ \end{align*}\] | \[\begin{align*} r=1 &\Rightarrow \text{perfect +ve linear correlation}\\ r=-1 &\Rightarrow \text{perfect -ve linear correlation}\\ r=0 &\Rightarrow \text{no linear correlation}\\ \end{align*}\] |
| On a histogram, \(\text{frequency density}=?\) | \[\text{f.d.}=\frac{\text{frequency}}{\text{class width}}\] |
| If \(Q_2-Q_1<Q_3-Q_2\), what type of skew does the data have? | Positive skew |
| If \(Q_2-Q_1>Q_3-Q_2\), what type of skew does the data have? | Negative skew |
| If \(Q_2-Q_1=Q_3-Q_2\), what type of distribution do we have? | A symmetrical distribution |
| If \(\text{mode}<\text{mean}<\text{median}\), what type of skew do we have? (This is true even if we only know 2 of mean, mode and median) | Positive skew |
| If \(\text{mode}=\text{mean}=\text{median}\), what type of distribution do we have? (This is true even if we only know 2 of mean, mode and median) | A symmetrical distribution |
| If \(\text{mode}>\text{mean}>\text{median}\), what type of skew do we have? (This is true even if we only know 2 of mean, mode and median) | Negative skew |
| How would you use the formula \(\frac{3(\text{mean}-\text{median})}{\text{standard deviation}}\) to determine how skewed some data are? | The closer the number is to zero the more symmetrical the data. The larger the number the greater the skew. A positive number implies positive skew. A negative number implies negative skew. |
| Which measures of location and dispersion are affected by extreme values? | Mean, standard deviation and range |
| Which measures of location and dispersion are NOT affected by extreme values? | Median and IQR |
| When comparing data sets, what 3 measures could you use in your comparison? | 1. A measure of location 2. A measure of dispersion 3. Skewness |
| What is meant by an independent (or explanatory) variable? | A variable that is set independently of the other variable. |
| What is meant by a dependent (or response) variable? | A variable whose values depend on the values of the independent variable. i.e. they are determined by the values of the independent variable |
| Is the product moment correlation coefficient affected by coded data? | No. \(r\) is not affected by coding. |
| For a discrete uniform distribution \(X\) defined over the values \(1, 2, 3, ..., n\), \[\text{E}(X)=?\] | \[E(X)=\frac{n+1}{2}\] |
| For a discrete uniform distribution \(X\) defined over the values \(1, 2, 3, ..., n\), \[\text{Var}(X)=?\] | \[\text{Var}(X)=\frac{n^2-1}{12}\] |
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