(a) Using the principle of mathematical induction, prove the inequality Σ for all n > 2. Show all your working. Σκk!

New Lottery (Questions 5-6) Suppose a new lottery randomly selects winners from players who buy a ticket and pays them $100 with probability p₁, $10 with probability p2, or $1 with probability P3. Losers get no money. The distribution of the amounts a player can win is shown as follows i1234 Wi $100 $10 $1 $0 Pi P1 P2 P3 P4 5. Multiple Choice: Which of the following distribution choices for probabilities P₁, P2, P3, P4 are a valid distribution for the amount won by a player. Distribution Choices A B с D E P1 P2 P3 P4 0.1 0.05 0.49 0.35 0.05 0.25 0.05 0.75 0.29 0.36 0.15 0.2 0.01 0.49 -0.050.55 0.17 0.33 0.14 0.36 6. Assume that the distribution of the amount won in the new lottery is as given below. Calculate the mean winning amount. ixi Pi 1$100 0.05 2 $10 0.10 3 $1 0.25 4 $0 0.60

Question 1133503: A CBS News poll conducted June 10 and 11, 2006, among a nationwide random sample of 651 adults, asked those adults about their party affiliation (Democrat, Republican or none) and their opinion of how the US economy was changing ("getting better," "getting worse" or "about the same"). The results are shown in the table below.**The column titles are (left to right above each of the number columns) are as follows: better, same, worseRepublican 38 104 44Democrat 12 87 137none 21 90 118Express each of your first five answers as a decimal and round to the nearest 0.001 (in other words, type 0.123, not 12.3% or 0.123456).(a) What fraction of survey respondents identified themselves as Republicans?(b) What fraction of survey respondents thought the economy was getting worse?

question 5