Q.6.20 - A First Course in Probability

Question

Let X1, X2, ... be a sequence of independent and identically distributed continuous random variables. Find 

a) $P\left\{X_6>X_1\mid X_1=\max \left(X_1,\dots ,X_5\right)\right\}$

b) $P\left\{X_6>X_2\mid X_1=\max \left(X_1,\dots ,X_5\right)\right\}$

Step-by-Step Solution

Answer

a) $P\left\{X_6>X_1\mid X_1=\max \left(X_1,\dots ,X_5\right)\right\}\text{ is }\frac{1}{6}$

b) $P\left\{X_6>X_1\mid X_1=\max \left(X_1,\dots ,X_5\right)\right\}=\frac{7}{12}$

Part (a) - Step 1: To determine

The value of $P\left\{X_6>X_1\mid X_1=\max \left(X_1,\dots ,X_5\right)\right\}$

Step 2: Explanation

Let $X_1,X_2.....$be continuous random variables with independent and identical distributions.

Conditional probability is defined as follows:

$\begin{array}{r}P\left\{X_6>X_1\mid X_1=max\left(X_1,\dots ,X_5\right)\right\}=\frac{P\left\{X_6>X_1,X_1=max\left(X_1,\dots ,X_5\right)\right\}}{P\left\{X_1=max\left(X_1,\dots ,X_5\right)\right\}}\\ =\frac{P\left\{\left(X_6>X_1\right)\cap \left[X_1=max\left(X_1,\dots ,X_5\right)\right]\right\}}{P\left\{X_1=max\left(X_1,\dots ,X_5\right)\right\}}\\ \approx \frac{P\left\{\left(X_6=max\left(X_1,\dots ,X_6\right)\right)\cap \left[X_1=max\left(X_1,\dots ,X_5\right)\right]\right\}}{P\left\{X_1=max\left(X_1,\dots ,X_5\right)\right\}}\\ \approx \frac{P\left(X_6=max\left(X_1,\dots ,X_6\right)\right)\times P\left(X_1=max\left(X_1,\dots ,X_5\right)\right)}{P\left\{X_1=max\left(X_1,\dots ,X_5\right)\right\}}\\ =\frac{\left(\frac{1}{6}\right)\left(\frac{1}{5}\right)}{\left(\frac{1}{5}\right)}\\ =5\cdot \left(\frac{1}{6}\times \frac{1}{5}\right)\\ =\frac{1}{6}\end{array}$

Hence $P\left\{X_6>X_1\mid X_1=\max \left(X_1,\dots ,X_5\right)\right\}\text{ is }\frac{1}{6}$

Part (b) - Step 3: To find

The value of $P\left\{X_6>X_2\mid X_1=\max \left(X_1,\dots ,X_5\right)\right\}$

Part (b) - Step 4: Explanation

Given: $P\left\{X_6>X_2\mid X_1=\max \left(X_1,\dots ,X_5\right),X_6>X_1\right\}$

Calculation: Let $X_1,X_2.....$be continuous random variables with independent and identical distributions.

Where $X_6>X_1$

$P\left\{X_6>X_2\mid X_1=\max \left(X_1,\dots ,X_5\right),X_6>X_1\right\}=1$

By symmetry,

$P\left\{X_6>X_2\mid X_1=\max \left(X_1,\dots ,X_5\right),X_6<X_1\right\}=\frac{1}{2}$

From part (a),

$P\left\{X_6>X_1\mid X_1=\max \left(X_1,\dots ,X_5\right)\right\}=\frac{1}{6}$

Therefore 

$\begin{array}{r}P\left\{X_6>X_2\mid X_1=max\left(X_1,\dots ,X_5\right)\right\}=\frac{1}{6}+\left(\frac{1}{2}\times \frac{5}{6}\right)\\ =\frac{7}{12}\end{array}$

Hence $P\left\{X_6>X_2\mid X_1=\max \left(X_1,\dots ,X_5\right)\right\} = \frac{7}{12}$