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Arithmetic sum
Show that
\[\sum_{j=1}^n j =\frac{1}{2}n(n+1)\]
Hint: notice that \(1+n=n+1\), \(2+(n-1)=n+1\), \(3+(n-2)=n+1\) and so on. How many such pairings exist ?
Coin Flips: describe events
Consider the probability space
\[ \Omega = \{ HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}\]
as the outcome of three consecutive tosses of a coin. (We make the reasonable assumption that all outcomes are are equally likely.) The event
\[ \{ HHH,TTT\}\]
is the event that all three tosses have the same outcome. Give a similar verbal description to each of the events bellow:
- \(\{HHH,HHT,HTH,HTT\}\)
- \(\{HTH,HTT,TTT,TTH\}\)
- \(\{HTT,HTH,HHT,HHH\}\)
- \(\{HTH,THH,TTH\}\)
- \(\{THT,HTT,TTH\}\)
- \(\{TTT,TTH,THT,HTT\}\)
- \(\{HHT,HHH,TTH,TTT\}\)
[Pitman, p31, #5] (Assign 1 and 5 first).
Calculus: Differentiation
Perform the following differentiation:
- \[\frac{d\ }{dx} \Big(x^4\Big) \]
- \[\frac{d\ }{dx} \Big(x^2 \exp(-x)\Big) \]
- \[\frac{d\ }{dx} \Big(\ln(x^2) \Big) \]
Calculus: Areas
Draw a picture of the region of the \(xy\)-plane were both x and y are between 0 and 1 and \(y \geq x^2\). Find the area of this region
Calculus : Exponentials integrals
Do the following integrals:
- \[ \int_0^1 x^3 dx\]
- (*) Hint: integrate by parts. \[\int_0^\infty x \exp(-x) dx\]
- (**) \[\int_0^\infty \exp(-\frac12 x^2) dx\] Try looking at \[\Big(\int_0^\infty \exp(-\frac12 x^2) dx\Big)\Big(\int_0^\infty \exp(-\frac12 y^2) dy\Big)=\int_0^\infty \int_0^\infty \exp(-\frac12 x^2) \exp(-\frac12 y^2)dx dy\] and then changing to polar coordinates. How des this help you figure out the original question ?
Practice with inclusion, exclusion.
Events \(A\), \(B\), and \(C\) are defined on a probability space. Find expressions for the
following probabilities in terms of \(\mathbf{P}(A)\), \(\mathbf{P}(B)\), \(\mathbf{P}(C)\), \(\mathbf{P}(AB)\), \(\mathbf{P}(AC)\), \(\mathbf{P}(BC)\), and \(\mathbf{P}(ABC)\).
- The probability that exactly two of the \(A\), \(B\), \(C\) occur.
- The probability that exactly one of these events occurs.
- The probability that none of these events occur.
Here the notation \(AB\) is short for \(A \cap B\) which is the event “both \(A\) and \(B\)”
( [Pitman, p. 31, # 10])
Random Letters
Suppose a word is picked at random from this sentence. Find:
- the outcome space for this random experiment
- the chance the word has at least 4 letters;
- the chance that the word contains at least 2 vowels (a,e,i,o,u)
- the chance that the word contains at least 4 letters and at least 2 vowels.
- What is the distribution of the length of the word picked ?
- What is the distribution of the number of vowels in the word ?
(Based on [Pitman p.9 #2 and p. 31 #6)
