IB Mathematics HL Year 2 Unit 7 (Core Topic 6: Probability and Statistics) Valuable Practice

Transcription

1 IB Mathematics HL Year 2 Unit 7 (Core Topic 6: Probability and Statistics) Valuable Practice 1. We have seen that the TI-83 calculator random number generator X = rand defines a uniformly-distributed random variable, where 0 X 1. Set Y = 10X 4. Using the fact that E(X) = 1 1 and VarX =, compute E(Y ) and Var(Y ) We have seen that the TI-83 calculator random number generator X = rand defines a uniformly-distributed random variable, where 0 X 1. Set Y = (2X 1) 2. (a) Using the fact that the density function for rand 2 looks like sketch the graph of the density function of Y. 1 (Make sure to label the axes clearly.) y x (b) Use your graph in part (a) to compute (or at least to estimate) E(Y ) and Var(Y ). E(Y ) = Var(Y) = 1 If you can t actually calculate this, can you give an educated guess?

2 3. Define the random variable X having density function f(x) = { a 1 x 2 if 1 x 1 0 otherwise. (a) Find a so that f really is a density function, and graph this density function. a = y x (b) Compute the mean of X. (c) Using the trig substitution x = sin θ or otherwise (possibly together with a suitable double-angle identity) compute the variance of X.

3 4. The probability density function f(x) of the continuous random variable X is defined on the interval [0, a] by (a) Find the value of a. f(x) = 1 x for 0 x 3, 8 27 for 3 < x a. 8x 2 (b) Find E(X) (c) Find Var(X).

4 5. (This is a modified version of problem 30L, # 7.) The lengths of steel rods produced by a machine are normally distributed with mean µ and standard deviation σ. It is known that 2% of all rods produced are less than 25 mm long and that 84% of all rods are less than 31 mm long. Compute µ and σ. µ = σ = 6. Suppose that an NBA basketball player has a field goal percentage of 44% (that is, on average, he makes 44% of his shots). In a given game he will take 20 shots. Let µ and σ be the mean and standard deviation of the number of shots X that he makes in a typical game. (a) Compute µ and σ. (b) Compute P(µ σ X µ + σ). µ = σ = (c) How does the probability computed in (b) compare with the theoretical probability for a normal distribution with the same mean and stardand deviation? (d) Compute the median of X.

5 7. Suppose that the number of injury accidents along a particularly dangerous stretch of road during a given year is a Poisson random variable X with mean 2.4. Assume next that some safety improvements were implemented along this road with the result that during the next year there was only one injury accident. Would you consider this to be indication that significant improvements in the safety of the road were made? Give an explanation. 8. Consider the Poisson random variable X with mean µ = 3. Therefore, P(X = n) = e 3 3 n /n!, n = 0, 1, 2,.... Let Y be the binomial random random variable for N = 100 trials and with p = Therefore P(Y = n) = ( ) 100 n 0.03 n n, and E(X) = E(Y ). Compare the distributions of X and Y by comparing P(X = n) and P(Y = n), n = 0, 1, 2,.... (Just do this using your calculator; do the probabilities seem to be approximately equal?) Finally, let Let Z be the binomial random random variable for N = 1000 trials and with p = Therefore P(Z = n) = ( ) 1000 n n n, and E(X) = E(Z). Which binomial random variable has a distribution closer to the Poisson random variable X, Y or Z? 2 2 There really is something going on here. It can be shown that for a given value µ (the expected value of a Poisson distribution) the limiting distribution (as N ) of the binomial distribution with N trials and probability p = µ/n approaches that of the Poisson distribution. A simple proof of this can be found at

6 9. Here s an extended problem. A geometric random variable is a random variable X having values 1, 2, 3,..., and whose distribution is given by a geometric sequence a, ar, ar 2,.... That is to say x n P (X = x) a ar ar 2 ar 3 ar n 1 (a) Find a so that as to make the above a discrete probability distribution. (b) Calculate E(X). Here s a hint as to how to proceed. Start by writing the geometric series with ratio x: Differentiate both sides: a + ax + ax 2 + ax 3 + = a + 2ax + 3ax 2 + 4ax 3 + = Does this help you compute E(X)? a, ( x < 1). 1 x a, ( x < 1). (1 x) 2 E(X) = (c) Use the same idea (compute the second derivative) to compute Var(X). Var(X) =