Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: Valid Comparisons and Informed Decisions: Probability Distributions, Exercise 10: Exercise 14B

When throwing a normal dice, let $X$ be the random variable defined by $X=$the square of the score shown on the dice. What is the expectation of $X$?

A "Fibonacci dice" is unbiased, six-sided and labelled with these numbers: $1, 2, 3, 5, 8, 13$. What is the expected score when the dice is rolled?

The discrete random variable X has probability distribution $P\left(x\right)=\frac{x}{36}$ for $x= 1, 2, 3,..... 8$. Find $E\left(X\right)$.

For the discrete random variable $X$, the probability distribution is given by

$P(X=x)=\left\{\begin{array}{ll}kx& x=1,2,3,4,5\\ k\left(10-x\right)& x=6,7,8,9\end{array}\right.$

Find the value of the constant $k$.

For the discrete random variable $X$, the probability distribution is given by

$P(X=x)=\left\{\begin{array}{ll}kx& x=1,2,3,4,5\\ k\left(10-x\right)& x=6,7,8,9\end{array}\right.$

Find $E\left(X\right)$.

Ten balls of identical size are in a bag. Two of the balls are red and the rest are blue. Balls are picked out at random from the bag and are not replaced. Let $R$ be the number of balls drawn out, up to and including the first red one. Calculate the mean value of $R$.

Ten balls of identical size are in a bag. Two of the balls are red and the rest are blue. Balls are picked out at random from the bag and are not replaced. Let $R$ be the number of balls drawn out, up to and including the first red one. What is the most likely value of $R$ ?

Consider the bag of balls in question $8$. Suppose now that each ball is replaced before the next is drawn. Calculate the probability that the first red is drawn after the third go.