X is a continuous random variable with probability density function f(x). The density is the curve y=x^2 for 0≤ x≤ 1, followed by the straight line running from the point (1; 1) down to the point (h; 0) on the x-axis, and f(x)=0 elsewhere. Write down the mode.

X is a continuous random variable with probability density function $f(x)$ . The density is the curve $y=x^{2}$ for $0\le x\le 1$ , followed by the straight line running from the point $(1;\,1)$ down to the point $(h;\,0)$ on the x-axis, and $f(x)=0$ elsewhere.

Write down the mode.

Model answer

Also accepted: 1,0, 1.0, x = 1, x=1

Explanation

The mode of a continuous random variable is the value of $x$ at which the density $f(x)$ is greatest.

On $0\le x\le 1$ the density climbs as $x^{2}$ , reaching $f(1)=1$ , and from $x=1$ onwards the straight line falls away to zero.

The peak is therefore at $x=1$ , so the mode is 1.

Derived from ZIMSEC Statistics Paper 4, June 2010, Q3

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