# Description of Probabilities and statistics

1. Let f(x) = 2e^(-(x-3)/c), 3 < x < infinity (zero otherwise) be a p.d.f. of a random variable X.

a. Find c

b. Find the CDF of X and sketch the CDF

c. Compute P(-5 < X < 10)

2. A candy maker produces mints that have a label weight of 30 grams. Assume that the distribution of the weights of these mints is N(30, 2^2).

a. Let X be the weight of a single mint selected at random from the production line. Find P(X > 32).

b. Suppose that 20 minutes are selected independently and weighted. Let Y equal the number of these mints that weigh less than 30 grams. Then find P(Y = 3).

c. Now, suppose that n = 100 mints are selected independently and weighted. Let Y equal the number of these mints that weigh less than 30 grams. Find the probability P(20 < Y <= 30) approximately. (Hint: Use CLT)

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#### Solution Preview

Solution to Q1.

(a) To find c, we need to use the property . So,

Integrating it, we have

i.e.,

i.e.,

2c=1

So, c=1/2.

(b) To find CDF, we need to compute F(x)= . We discuss two ...

#### Solution Summary

The probabilities and statistics for a random variable is determined.