Unformatted text preview: 51 cards, and we observe that this card is a diamond. Given
this information, ﬁnd the probability that the lost card is a club. b. Four cards are selected at random without replacement from the 51 cards, and we observe that all
are hearts. Given this information, ﬁnd the probability that the lost card is a club. 13 Example 7
Suppose that P (X = 0) = 1 − P (X = 1). If E (X ) = 3Var(X ), ﬁnd:
a. P (X = 0). b. V ar(3X ).
Example 8
Let X ∼ b(n, p). Show that V ar(X ) = np(1 − p). Hint: First ﬁnd E [(X (X − 1)] and then use σ 2 = EX 2 − µ2 .
Example 9
Let X be a geometric random variable with probability of success p. The probability mass function of X is:
P (X = k ) = (1 − p)k−1 p, k = 1, 2, · · · Show that the probabilities sum up to 1.
Example 10
In a game of darts, the probability that a particular player aims and hits treble twenty with one dart is 0.40.
How many throws are necessary so that the probability of hitting the treble twenty at least once exceeds 90%?
Example 11
Show that the mean and variance of the hypergeometric distribution are:
µ= nr
N σ2 = nr(N − r)(N − n)
N 2 (N − 1) Example 12
Using steps similar to those employed to ﬁnd the mean of the binomial distribution show that the mean of the
r
negative binomial distribution is µ = p . Also show that the variance of the negative binomial distribution
r1
2
is σ = p ( p − 1) by ﬁrst evaluating E [X (X + 1)].
Example 13
r
Show that if we let p = N , the mean and the variance of the hypergeometric distribution can be written as
2
µ = np and σ = np(1 − p) N −n . What does this remind you?
N −1
Example 14
Suppose 5 cards are selected at random and without replacement from an ordinary deck of playing cars.
a. Construct the probability distribution of X , the number of clubs among the ﬁve cards.
b. Use (a) to ﬁnd P (X ≤ 1). Example 15
The manager of an industrial plant is planning to buy a new machine of either type A, or type B . The
number of daily repairs XA required to maintain a machine of type A is a random variable with mean and
variance both equal to 0.10t, where t denotes the number of hours of daily operation. The number of daily
repairs XB for a machine of type B is a random variable with mean and variance both equal 0.12t. The daily
2
2
cost of operating A is CA (t) = 10t + 30XA , and for B is CB (t) = 8t + 30XB . Assume that the repairs take
negligible time and that each night the machines are tuned so that they operate essentially like new machines
at the start of the next day. Which machine minimizes the expected daily cost if a workday consists of
a. 10 hours
b. 20 hours 14 Discrete random variables  some examples (solutions)
Example 1
Let X be the payoﬀ:
Card X ($)
P (X )
4
Jack
15
52
4
Queen
15
52
4
King
5
52
4
Ace
5
52
36
Else
4
52
Therefore the expected gain is:
4
4
4
4
E (X ) = 15 52 + 15 52 + 5 52 + 5 52 − 4 36 = $ 16 .
52
52
Example 2
This is hypergeometric problem. The probability of accepting the lot is the pr...
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This note was uploaded on 10/04/2012 for the course STATISTICS 100a taught by Professor Cristou during the Spring '10 term at UCLA.
 Spring '10
 CRISTOU
 Binomial

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