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Probability density functions practice problems

Normalize a probability density, then find a probability and the mean. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

8 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 5.592

medium
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- 2 x}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 2) \) and the mean of \( \displaystyle X \).
  1. A density integrates to 1 over its whole range.
    Reviewed
  2. \[ \int\limits_{0}^{\infty} e^{- 2 x}\, dx = \frac{1}{2} \]
    The integral without k.✓ Proved
  3. \[ 2 \cdot 1 \cdot \frac{1}{2} = 1 \]
    So k = 2: k times that integral is 1.✓ Proved
  4. \[ \int\limits_{0}^{2} 2 e^{- 2 x}\, dx = 1 - e^{-4} \]
    The probability is the area over the interval.✓ Proved
  5. \[ \int\limits_{0}^{\infty} 2 x e^{- 2 x}\, dx = \frac{1}{2} \]
    The mean, ∫ x f(x) dx.✓ Proved
Answer \( k = 2,\ P = 1 - e^{-4},\ \mu = \frac{1}{2} \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.595

medium
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(1 - x\right) \) for \( \displaystyle 0 \le x \le 1 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le 1) \) and the mean of \( \displaystyle X \).
  1. A density integrates to 1 over its whole range.
    Reviewed
  2. \[ \int\limits_{0}^{1} \left(1 - x\right)\, dx = \frac{1}{2} \]
    The integral without k.✓ Proved
  3. \[ 2 \cdot 1 \cdot \frac{1}{2} = 1 \]
    So k = 2: k times that integral is 1.✓ Proved
  4. \[ \int\limits_{\frac{1}{2}}^{1} \left(2 - 2 x\right)\, dx = \frac{1}{4} \]
    The probability is the area over the interval.✓ Proved
  5. \[ \int\limits_{0}^{1} 2 x \left(1 - x\right)\, dx = \frac{1}{3} \]
    The mean, ∫ x f(x) dx.✓ Proved
Answer \( k = 2,\ P = \frac{1}{4},\ \mu = \frac{1}{3} \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.596

medium
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x^{2}\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le 4) \) and the mean of \( \displaystyle X \).
  1. A density integrates to 1 over its whole range.
    Reviewed
  2. \[ \int\limits_{0}^{4} x^{2}\, dx = \frac{64}{3} \]
    The integral without k.✓ Proved
  3. \[ 3 \cdot \frac{1}{64} \cdot 64 \cdot \frac{1}{3} = 1 \]
    So k = 3/64: k times that integral is 1.✓ Proved
  4. \[ \int\limits_{\frac{1}{2}}^{4} \frac{3 x^{2}}{64}\, dx = \frac{511}{512} \]
    The probability is the area over the interval.✓ Proved
  5. \[ \int\limits_{0}^{4} \frac{3 x^{3}}{64}\, dx = 3 \]
    The mean, ∫ x f(x) dx.✓ Proved
Answer \( k = \frac{3}{64},\ P = \frac{511}{512},\ \mu = 3 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.597

medium
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- \frac{x}{2}}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 2) \) and the mean of \( \displaystyle X \).
  1. A density integrates to 1 over its whole range.
    Reviewed
  2. \[ \int\limits_{0}^{\infty} e^{- \frac{x}{2}}\, dx = 2 \]
    The integral without k.✓ Proved
  3. \[ 1 \cdot \frac{1}{2} \cdot 2 = 1 \]
    So k = 1/2: k times that integral is 1.✓ Proved
  4. \[ \int\limits_{0}^{2} \frac{e^{- \frac{x}{2}}}{2}\, dx = 1 - e^{-1} \]
    The probability is the area over the interval.✓ Proved
  5. \[ \int\limits_{0}^{\infty} \frac{x e^{- \frac{x}{2}}}{2}\, dx = 2 \]
    The mean, ∫ x f(x) dx.✓ Proved
Answer \( k = \frac{1}{2},\ P = 1 - e^{-1},\ \mu = 2 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.600

medium
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 2) \) and the mean of \( \displaystyle X \).
  1. A density integrates to 1 over its whole range.
    Reviewed
  2. \[ \int\limits_{0}^{2} x\, dx = 2 \]
    The integral without k.✓ Proved
  3. \[ 1 \cdot \frac{1}{2} \cdot 2 = 1 \]
    So k = 1/2: k times that integral is 1.✓ Proved
  4. \[ \int\limits_{0}^{2} \frac{x}{2}\, dx = 1 \]
    The probability is the area over the interval.✓ Proved
  5. \[ \int\limits_{0}^{2} \frac{x^{2}}{2}\, dx = \frac{4}{3} \]
    The mean, ∫ x f(x) dx.✓ Proved
Answer \( k = \frac{1}{2},\ P = 1,\ \mu = \frac{4}{3} \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.602

medium
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- x}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 1) \) and the mean of \( \displaystyle X \).
  1. A density integrates to 1 over its whole range.
    Reviewed
  2. \[ \int\limits_{0}^{\infty} e^{- x}\, dx = 1 \]
    The integral without k.✓ Proved
  3. \[ 1 \cdot 1 = 1 \]
    So k = 1: k times that integral is 1.✓ Proved
  4. \[ \int\limits_{0}^{1} e^{- x}\, dx = 1 - e^{-1} \]
    The probability is the area over the interval.✓ Proved
  5. \[ \int\limits_{0}^{\infty} x e^{- x}\, dx = 1 \]
    The mean, ∫ x f(x) dx.✓ Proved
Answer \( k = 1,\ P = 1 - e^{-1},\ \mu = 1 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.603

medium
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(1 - x\right)\right) \) for \( \displaystyle 0 \le x \le 1 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 1) \) and the mean of \( \displaystyle X \).
  1. A density integrates to 1 over its whole range.
    Reviewed
  2. \[ \int\limits_{0}^{1} x \left(1 - x\right)\, dx = \frac{1}{6} \]
    The integral without k.✓ Proved
  3. \[ 6 \cdot 1 \cdot \frac{1}{6} = 1 \]
    So k = 6: k times that integral is 1.✓ Proved
  4. \[ \int\limits_{0}^{1} 6 x \left(1 - x\right)\, dx = 1 \]
    The probability is the area over the interval.✓ Proved
  5. \[ \int\limits_{0}^{1} 6 x^{2} \left(1 - x\right)\, dx = \frac{1}{2} \]
    The mean, ∫ x f(x) dx.✓ Proved
Answer \( k = 6,\ P = 1,\ \mu = \frac{1}{2} \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.604

medium
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{3}{2} \le X \le \frac{5}{2}) \) and the mean of \( \displaystyle X \).
  1. A density integrates to 1 over its whole range.
    Reviewed
  2. \[ \int\limits_{0}^{3} \left(x + 1\right)\, dx = \frac{15}{2} \]
    The integral without k.✓ Proved
  3. \[ 2 \cdot \frac{1}{15} \cdot 15 \cdot \frac{1}{2} = 1 \]
    So k = 2/15: k times that integral is 1.✓ Proved
  4. \[ \int\limits_{\frac{3}{2}}^{\frac{5}{2}} \left(\frac{2 x}{15} + \frac{2}{15}\right)\, dx = \frac{2}{5} \]
    The probability is the area over the interval.✓ Proved
  5. \[ \int\limits_{0}^{3} \frac{2 x \left(x + 1\right)}{15}\, dx = \frac{9}{5} \]
    The mean, ∫ x f(x) dx.✓ Proved
Answer \( k = \frac{2}{15},\ P = \frac{2}{5},\ \mu = \frac{9}{5} \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More probability density functions practice

Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(1 \le X \le \frac{5}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.593medium✓ Every equation proved
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- 2 x}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.594medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x^{2}\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le \frac{3}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.598medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(2 \le X \le \frac{5}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.599medium✓ Every equation proved
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x^{2}\right) \) for \( \displaystyle 0 \le x \le 1 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.601medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- \frac{x}{3}}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le \frac{1}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.605medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- \frac{x}{3}}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.606medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- 3 x}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 2) \) and the mean of \( \displaystyle X \).
Problem 5.657medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x\right) \) for \( \displaystyle 0 \le x \le 1 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.658medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 1 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.659medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.660medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(3 - x\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{5}{2} \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.661medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(4 - x\right)\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(1 \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.662medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(4 - x\right)\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le 2) \) and the mean of \( \displaystyle X \).
Problem 5.663medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.664medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(2 \le X \le \frac{5}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.665medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x^{2}\right) \) for \( \displaystyle 0 \le x \le 1 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.666medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- x}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le \frac{1}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.667medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(3 - x\right)\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.668medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le \frac{5}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.669medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{3}{2} \le X \le \frac{5}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.670medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(1 \le X \le \frac{3}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.671medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(2 - x\right)\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.672medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(1 - x\right)\right) \) for \( \displaystyle 0 \le x \le 1 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le \frac{1}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.673medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(2 \le X \le \frac{5}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.674medium✓ Every equation proved
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(3 - x\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(1 \le X \le \frac{3}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.675medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.676medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x\right) \) for \( \displaystyle 0 \le x \le 1 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.677medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(3 - x\right)\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(1 \le X \le 2) \) and the mean of \( \displaystyle X \).
Problem 5.678medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x^{2}\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(1 \le X \le \frac{3}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.679medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(2 - x\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(1 \le X \le \frac{3}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.680medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x^{2}\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.681medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- 2 x}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le \frac{1}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.703medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(4 - x\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{3}{2} \le X \le 4) \) and the mean of \( \displaystyle X \).
Problem 5.704medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- \frac{x}{3}}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 2) \) and the mean of \( \displaystyle X \).
Problem 5.705medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x^{2}\right) \) for \( \displaystyle 0 \le x \le 1 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le \frac{1}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.706medium✓ Every equation proved
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(2 - x\right)\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 2) \) and the mean of \( \displaystyle X \).
Problem 5.707medium✓ Every equation proved
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(3 - x\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(1 \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.708medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x^{2}\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{5}{2} \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.709medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(4 - x\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(2 \le X \le 4) \) and the mean of \( \displaystyle X \).
Problem 5.710medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(4 - x\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{3}{2} \le X \le \frac{5}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.711medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- 3 x}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 1) \) and the mean of \( \displaystyle X \).
Problem 5.712medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- 3 x}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.773medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(4 - x\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(3 \le X \le 4) \) and the mean of \( \displaystyle X \).
Problem 5.774medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{3}{2} \le X \le 2) \) and the mean of \( \displaystyle X \).
Problem 5.775medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(2 - x\right)\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{1}{2} \le X \le 2) \) and the mean of \( \displaystyle X \).
Problem 5.776medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x^{2}\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{5}{2} \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.777medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x\right) \) for \( \displaystyle 0 \le x \le 3 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(1 \le X \le \frac{3}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.778medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x\right) \) for \( \displaystyle 0 \le x \le 4 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(\frac{5}{2} \le X \le \frac{7}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.779medium✓ Every equation proved
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- x}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 2) \) and the mean of \( \displaystyle X \).
Problem 5.780medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x + 1\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le \frac{3}{2}) \) and the mean of \( \displaystyle X \).
Problem 5.781medium✓ Nihil obstat
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(e^{- \frac{x}{2}}\right) \) for \( \displaystyle x \ge 0 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 3) \) and the mean of \( \displaystyle X \).
Problem 5.782medium✓ Nihil obstat