Exercise Set 16

  1. You throw five fair dice and count the number of sixes you get. Let \(X\) be the number of sixes. Tabulate \(P(X=x)\) for \(x=0, 1,\dots,5\).

  2. A random variable \(X\) has Poisson distribution with rate parameter \(\lambda=2\). Find:

    1. \(P(X \le 2)\)
    2. \(P(X=3)\)
    3. \(P(X \ge 4)\)

  3. A discrete random variable \(X\) takes values in the set \(\{1,2,3,4\}\) and has \(P(X=x)=x/10\) for each \(x\) in this set. Find:

    1. \(\operatorname{E}(X)\)
    2. \(\operatorname{E}(X^2)\)
    3. \(\operatorname{var}(X)\)

  4. A discrete random variable \(X\) takes values in the set \(\{1,2,3\}\) and has \(P(X=x)=c/x\) for each \(x\) in this set, where \(c\) is some suitable constant. Find \(c\) and so find

    1. \(\operatorname{E}(X)\)
    2. \(\operatorname{E}(X^2)\)
    3. \(\operatorname{var}(X)\)

  5. You throw two fair dice. Let \(X\) denote the sum of the two numbers thrown. Find:

    1. \(\operatorname{E}(X)\)
    2. \(\operatorname{E}(X^2)\)
    3. \(\operatorname{var}(X)\)

  6. 2% of packages produced by a packaging machine have defective seals. What is the probability that a batch will contain more than 2 defective packages if the batch size is (a) 20 (b) 50?

  7. A rifle range competitor scores one hit in every 4 shots, on the average. Assuming that the binomial distribution is applicable, if they have four shots in a session what is:

    1. the probability that they will get exactly one hit?
    2. the probability that they will get at least one hit?

  8. 2% of cans leaving a canning factory have a defective paint finish. Cans are packed in boxes in groups of 50. Determine the mean and standard deviation of the number of cans in a box which have a defective finish.

  9. Sheets of metal have a plating fault which occurs randomly at an average rate of 1 per \(\text{m}^2\). What is the probability that a sheet \(1.5\text{m} \times 2 \text{m}\) will have:

    1. At most one fault?
    2. At least one fault?

  10. 250 litres of water have been polluted with \(10^6\) bacteria. What is the probability that a sample of 1 ml of the water contains no bacteria?

  11. An average of 264 vehicles an hour pass along a stretch of road each taking 30 seconds to travel along it. What is the probability that at a given instant there will be no vehicles in the road?