Suppose that three fair 6-sided dice are rolled.

- Let \(M\) be the minimum of three numbers rolled. Find \(\mathbb{E}(M)\).
- Let \(S\) be the sum of the largest two rolls. Find \(\mathbb{E}(S)\).

Learning probability by doing !

Suppose that three fair 6-sided dice are rolled.

- Let \(M\) be the minimum of three numbers rolled. Find \(\mathbb{E}(M)\).
- Let \(S\) be the sum of the largest two rolls. Find \(\mathbb{E}(S)\).

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Posted in Dice Rolls, Expectations, Max and Mins, Tail Sum Fromula

Let \(U_1,U_2,U_3,U_4,U_5\) be independent, each with uiform distribution on \((0,1)\). Let \(R\) be the distance between the max and the min of the \(U_i\)’s. Find

- \(\mathbf{E} R\)
- the joint density of the max and the min of the \(U_i\)’s.
- the \(\mathbf{P}(R> .5)\)

[pitman p355, #14]

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Posted in Max and Mins, Order Statistics

Suppose \(U_1,U_2, …\) are independent uniform \( (0,1) \) random variables. Let \(N\) be the first point of increase. That is to say the first \(n \geq 2\) such that \(U_n > U_{n-1}\). Show that for \(u \in (0,1)\):

- \[\mathbf{P}(U_1 \leq u \ { and } \ N=n)= \frac{u^{n-1}}{(n-1)!}-\frac{u^{n}}{n!} \quad;\quad n \geq 2\]
- \( \mathbf{E}(N)=e \)

Some useful observations:

- \[\mathbf{P}(U_1 \leq u \ { and } \ N=n) = \mathbf{P}(U_1 \leq u \ { and } \ N \geq n) -\mathbf{P}(U_1 \leq u \ { and } \ N \geq n+1)\]
- The following events are equal

\[ \{U_1 \leq u \quad{ and } \quad N \geq n\} = \{U_{n-1}\leq U_{n-2} \leq \cdots \leq U_2\leq U_{1} \leq u \}\] - \[ \mathbf{P}\{U_2 \leq U_1 \leq u \}= \int_0^u \int_0^{u_1} du_2 du_1 \]

Suppose \(R_1\) and \(R_2\) are two independent random variables with the same density function

\[f(x)=x\exp(-{\textstyle \frac12 }x^2)\]

for \(x\geq 0\). Find

- the density of \(Y=\min(R_1,R_2)\);
- the density of \(Y^2\)

[Pitman p. 336 #21]

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Posted in Basic probability, Change of Variable, Max and Mins

Tagged JCM_math230_HW8_S13, JCM_math230_HW9_F22, JCM_math230_HW9_S15, JCM_math340_HW8_F13

Let \(X_1\) and \(X_2\) be random variables and let \(M=\mathrm{max}(X_1,X_2)\) and \(N=\mathrm{min}(X_1,X_2)\).

- Argue that the event \(\{ M \leq x\}\) is the same as the event \(\{X_1 \leq x, X_2 \leq x\}\) and similarly that t the event \(\{ N > x\}\) is the same as the event \(\{X_1 > x, X_2 > x\}\).
- Now assume that the \(X_1\) and \(X_2\) are independent and distributed with c.d.f. \(F_1(x)\) and \(F_2(x)\) respectively . Find the c.d.f. of \(M\) and the c.d.f. of \(N\) using the proceeding observation.
- Now assume that \(X_1\) and \(X_2\) are independently and exponentially distributed with parameters \(\lambda_1\) and \(\lambda_2\) respectively. Show that \(N\) is distributed exponentially and identify the parameter in the exponential distribution of \(N\).
- The route to a certain remote island contains 4 bridges. If the time to collapse of each bridge is exponential distributed with mean 20 years and is independent of the other bridges, what is the distribution of the time until the road is impassable because one of the bridges has collapsed.

[Jonathan Mattingly]

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Posted in cumulative distribution function, Exponential Random Variables, Max and Mins

Tagged JCM_math230_HW7_S13, JCM_math230_HW9_F22, JCM_math230_HW9_S15, JCM_math340_HW7_F13

Suppose \(X_1, … , X_{17}\) are iid uniform on \( (.5,.8) \). What is \({\mathbf{E}} [X_{(k)}] \) ?

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Posted in Max and Mins, Order Statistics

Suppose \(X_1, … , X_n \stackrel{iid}{\sim} U(0,1) \). How large must \(n\) be to have that \({\mathbf{P}}(X_{(n)} \geq .95) \geq 1/2\) ?

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Posted in Max and Mins, Order Statistics

Tagged JCM_math230_HW10_F22, JCM_math230_HW9_S15, JCM_math340_HW7_F13

Let \(X_1,…,X_n \stackrel{iid}{\sim} \mbox{Exp}(\lambda) \)

Let \(V = \mbox{min}(X_1,…,X_n)\) and \(W = \mbox{max}(X_1,…,X_n)\).

What is the joint distribution of \(V,W\). Are they independent ?

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Posted in Joint Distributions, Max and Mins, Order Statistics

Let \(X_1,\cdots, X_n\) be random variables which are *i.i.d.* \(\text{unifom}(0,1)\). Let \(X_{(1)},\cdots, X_{(n)}\) be the associated order statistics.

- Find the distribution of \(X_{(n/2)}\) when \(n\) is even.
- Find \(\mathbf{E} [ X_{(n)} – X_{(1)} ]\).
- Find the distribution of \(R=X_{(n)} – X_{(1)}\).

There are \(15\) stock brokers. The returns (in thousands of dollars) on each brokers is modeled as a separate independent exponential distribution \(X_1 \sim \mbox{Exp}(\lambda_1),…,X_{15} \sim \mbox{Exp}(\lambda_{15})\). Define \(Z = \min\{X_1,…,X_{15}\}\).

What is \(\mathbf{E}(Z)\) ?

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Posted in Expectations, Exponential Random Variables, Max and Mins

Let \(X_1,…,X_5\) be five iid rolls of six sided die. Let \(Z = \mbox{max}\{X_1,…,X_5\}\). Compute \(\mathbf{E}(Z)\).

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Posted in Dice Rolls, Max and Mins

Two dice are rolled. Find the probabilities of the following events.

a) the maximum of the two numbers rolled is less than or equal to 2;

b) the maxinum of the two numbers rolled is less than or equal to 3;

c) the maximum of the two numbers rolled is exactly equal to 3;

d) Repeat b) and c) with 3 replaced by \(x=1,…,6\);

e) Denote \( \mathbf{P}(x)\) as the probability that the maximum number is exactly \(x\).

Compute \( \sum_{x=1}^6\mathbf{P}(x)\).

[Pitman Page 10, #7]

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Posted in Dice Rolls, Max and Mins, Sequence of independent trials

Let \(X_1\) and \(X_2\) be the number obtained on two rolls of a fair die. Let \(Y_1=\max(X_1,X_2)\) and \(Y_2=\min(X_1,X_2)\).

- Display the joint distribution tables for \( (X_1,X_2)\).
- Display the joint distribution tables for \( (Y_1,Y_2)\).
- Find the distribution of \(X_1X_2\).

Combination of [Pitman, p. 159 #4 and #5]

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Posted in Dice Rolls, Max and Mins

Tagged JCM_math230_HW4_S13, JCM_math230_HW56_F22, JCM_math230_HW5_S15, JCM_math340_HW4_F13

Suppose that a die has \(n\) sides. Compute the probability that:

- the maximum of the two numbers rolled is less than or equal to 2;
- the maximum of the two numbers rolled is less than or equal to \( i \in \{1,\dots, n\} \);
- the maximum of the two numbers rolled is equal to \( i \in \{1,\dots,n\}\).

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Posted in Basic probability, Max and Mins

Suppose two 4-sided dice are rolled. Find the probabilities of the following events:

- the maximum of the two numbers rolled is less than or equal to 2;
- the maximum of the two numbers rolled is less than or equal to 3;
- the maximum of the two numbers rolled is equal to 3;
- Repeat part c for the maximum equal to 1, 2, and 4.
- If M is the maximum of the two numbers, then

\[\mathbf{P}(M = 1) + \mathbf{P}(M = 2) + \mathbf{P}(M = 3) + \mathbf{P}(M = 4) = 1 \]

check that your answers for 3) and 5) satisfy this relationship.

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Posted in Basic probability, Max and Mins