ANSWER ALL THE QUESTIONS. 1. 10 points Given X B(8,0.4),...

ANSWER ALL THE QUESTIONS. 1. 10 points Given X ~ B(8,0.4), find P(X > 1) 2. 10 points Given X ~ Po(2), the probability of P(X < 2) is 5e?² True False 3. 10 points Given X ~ N(10,36) find the P(7 < X < 16) 4. 15 points If X ~ N(?, ?²) find the value of ? given that P(X < 400) = 0.1587 and P(X > 500) = 0.0668. 5. 5 points Given X ~ B(5,0.4), find Var(x) Figure 3.3.1: A sketch of the transformation between intervals x ? [0, 2?] and t ? [0, L]. Recall the form of the Fourier representation for f(x) in Equation (3.2.1): f(x) ~ a?/2 + ????? [a? cos nx + b? sin nx]. (3.3.1) Inserting the transformation relating x and t, we have g(t) ~ a?/2 +

Transcript

00:01 Here we have five questions.

00:02 The first is given a binomial random variable x with parameter n equals 8 and p equals 0 .4.

00:10 Find the probability that x is greater than 1.

00:15 We want the probability that x is greater than 1.

00:19 This can be re -expressed as 1 minus probability that x is at most 1.

00:24 The probability function for the binomial random variable is given by this formula.

00:46 So we can solve the probability that x is at most 1 as the probability that x equals 0 plus the probability that x equals 1.

00:55 For x equals 0, the probability function simplifies to 0 .6 to the exponent 8.

01:03 And for x equals 1, we have h choose 1 times 0 .4, which is 0 .6 to the exponent 7.

01:20 And this comes out to approximately 0 .8936.

01:24 For the second question, we consider a random variable that follows a poisson distribution distribution with a mean of 2 and we're asked true or false.

01:40 The probability that x is less than 2 is 5 times e to the minus 2.

01:52 Followed up the probability function for a poisson random variable is as follows.

01:58 The probability that x is less than 2, the probability that x is at most 1.

02:17 This is the probability of x equals 0 plus the probability that x equals equals 1.

02:23 For x equals 0, the probability function simplifies to e to the minus mu.

02:29 In this case, mu is 2.

02:34 For x equals 1, we have 2 times e to the minus 2.

02:39 This comes out to 3 times e to the minus mu, which is not 5 times e to the minus 2.

02:54 The statement is false.

02:55 For question 3, we consider a random variable that is normally distributed with with a mean of 10 and a variance of 36.

03:18 We're asked to find the probability that x is between seven and 16.

03:27 Let's first express this in terms of cumulative probabilities.

03:29 It's equal to the probability that x is less than 16 minus the probability that x is at most seven.

03:39 We can standardize the random variable according to this formula so that we can use a standard normal table.

03:46 Doing so, we have the probability that z is less than one minus the probability that z is at most minus 0 .5.

04:02 We can look up z equals 1 in a standard normal table.

04:04 It corresponds to a cumulative probability 0 .8413.

04:17 Look up z equals minus 0 .5 in the table, and we get 0 .3085...

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